Odds & Probability in Blackjack. enum Coin { Fair, DoubleHeaded } Now let's suppose we have a bag of coins. Again, draw a probability tree. Find the probability that the coin is heads. It comes up heads each time. What is the chance of getting two heads? Easy, it's 0. When one of the coins is randomly selected and ﬂipped, it shows heads. Instead of probability distributions, we use probability densities and integrate over ranges of possible. Changing either H or E can change the value of the ip. Online virtual coin toss simulation app. Solution: a) A tree diagram of all possible outcomes. What is the probability that two headed coin was selected? Denote with Ak the event that randomly selected coin lands heads up k times. You randomly take one of them out of your pocket without looking at it. Pick one coin at random and flip it. Two-Face said that heads he would free Batman, tails he would kill, and Batman asked about the edge, and said he should agree to turn himself in and co-operate with all plastic surgery and. We begin with some previews. At the root (level 0) we choose randomly the first coin. If heads turn up each time. (a)A gambler has a fair coin and a two-headed coin in his pocket. 3, 6 There are three coins. But the coin comes up heads 8 times in a row. According to Shannon, the information content of this message is zero. One of the three coins is chosen at random and tossed, and it shows heads. The Coin Toss Probability Calculator an online tool which shows Coin Toss Probability for the given input. Find the pr Algebra -> Probability-and-statistics -> SOLUTION: A box contains 3 coins: one coin with two sides-head & tail, one coin two headed and one coin with probability of heads is 1/3. Byju's Coin Toss Probability Calculator is a tool which makes calculations very simple and interesting. Given that the coin is heads, find the conditional probability of each coin type. Theoretical and experimental probability: Coin flips and die rolls. The hypotheses areH1-the coin is two headed, and H2 the coin is fair. We use the experiement of tossing a coin three times to create the probability distribution table for the number of heads. Now what is the probability that it is a fair coin?. Random number list to run experiment. The answer is 1/2. Consider the null hypothesis that a coin is 2-headed (prob of heads is 1) and the observed data of 1 flip of the coin which came up heads. But the coin comes up heads 8 times in a row. So far we’ve established that: The probability of flipping heads or tails is equally likely each individual toss: P(H) = P(T) = 1/2. Signiﬁcance: among 1000 coins, if one comes up heads 10 times in a row, is it likely to be a 2-headed coin? Applications to economics, investment and hiring. There are 3 coins in a box. You have two coins in your pocket. The question is: A bag contains 6 coins, 2 of which have a head on both sides while the other 4 coins are normal. What is the probability that the 2-headed coin is chosen? To solve this, note that Pf2-headed coing= 1 2; Pffair coing= 1 2: and Pfn headsj2-headed coing= 1; Pfn headsjfair coing= 2 n: By thelaw of total probability, Pfn headsg = Pfn headsj2-headed. “The coin tosses are independent events; the coin doesn’t have a memory. There is a probability of 0. At level 1 we toss it. One is a two-headed coin (having a head on both faces), another is a biased coin that comes up heads 75% of the times, and third is also a biased coin that comes up tails 40% of the times. One is a two-headed coin ( having head on both faces ), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. Prior and posterior beliefs are assessments of probability before and after seeing an outcome. What is the chance of getting two heads? Easy, it's 0. This is, however, wrong, because given that heads came up, it is more likely that the two-headed coin was chosen. You have a jar containing 999 fair coins and one two-headed coin. ” A coin is selected at random, ﬂipped n times and in all ﬂips it falls heads up. I've read that if you flip a coin 10 times and it comes up heads every time then it's still a 50/50 chance to be a heads or tails on the 11th flip. Alice tosses this coin repeatedly. What is the conditional probability that. One coin is selected from the box at random and the face of one side is observed. Now, the probability of getting three heads on three throws of a regular coin is $1/8$, as you can surely check. You know that the coins you are given to test are either unfair coins with heads probability 1 4; unfair coins with heads probability 3 4; and fair coins. Bag containing 3 coins probability question? A bag contains 3 coins, one double headed, one fair and a biased one in such that heads comes up with a probability of 3/4. When one of the three coins is selected at random and ﬂipped, it shows heads. Let H 1 first coin flip is heads H 2 second coin flip is heads The likelihood of a coin flip coming up heads is 0. If heads appears both times, what is the probability that the coin is two-head. Similar Questions. There are 3 coins, 2 fair and 1 is a two headed coin. a) What is the probability that the coin chosen is the two-headed coin?. One is a two headed coin (having head on both faces),another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. b) The probability of getting blue on the spinner and head on the coin. I think hugin is right: the probability of a 6th head is just the combination of the probability you have the double-headed coin (0. The second one is a fair coin. Chapter 3 Probability contrast, if we flip a two-headed coin we do know for sure what the result will be, heads! An experiment in which any one of number of possible outcomes may result is called a random experiment or probability experiment. You flip it and it comes up "heads". Given that the coin comes up heads, what is the probability that the fair coin was ipped? 3. Conditional probability. For a two-headed coin, we might have P (X =H)=1 and P (X =T)=0. You draw the two-headed coin and see face 2. What is the probability that it is the fair coin? (b) Suppose that he flips the same coin a second time and, again, it shows heads. Your roommate has the two-headed coin and a regular two-sided fair coin in their pocket. Every time the result of the tossing is head. When one of the 3 coins is selected at random and ipped, it shows heads. When one of the coins is randomly selected and ﬂipped, it shows heads. Two views of belief 297 Since the ratio of these probabilities is 2" 1, the belief/probability function. , in short (H, H) or (H, T) or (T, T) respectively; where H is denoted for head and T is denoted for tail. I think hugin is right: the probability of a 6th head is just the combination of the probability you have the double-headed coin (0. 3 Q6 There are three coins. 5 is the probability that the selected coin is a fair coin. Case 2: One head. In my town, it's rainy one third of the days. Bag containing 3 coins probability question? A bag contains 3 coins, one double headed, one fair and a biased one in such that heads comes up with a probability of 3/4. Now flip that coin three times. What is the probability that it is the fair coin?. What is now the probability that it is the fair coin?. One coin has heads on both sides, one coin has tails on both sides, the third one has head on one side and tail on the other side. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. There are three coins in a box. Question 266203: two coin and one six sided number cube are tossed together. GMAT Math: the Probability "At Least" Question By Mike MᶜGarry on December 19, 2012 , UPDATED ON April 20, 2019, in GMAT Math , GMAT Math Basics In the first post in this series, I spoke about the AND rule and the OR rule in probability. A box has a 2-headed coin and a fair coin. When one of the 3 coins is selected at random and. A jar has 1000 coins, of which 999 are fair and 1 is double headed. (a) You pick a coin at random and toss it. Your friend takes a coin out of his pocket and tosses it three times. Three fair dice are thrown. A two-headed coin is a coin which has head on both sides; a fair coin means it has tail on one side and head on the other. What is the chance of getting two heads? Easy, it's 0. on each side. If you toss a coin, it will come up a head or a tail. Mint has made the following coins: A unique double-headed Jefferson nickel struck with 2 obverse dies. At level 1 we toss it. The probability of Heads for the first coin is 1/3, and the probability of Heads for the second is 2/3. One is a two-headed coin, another is a fair coin, and the third is a biased coin that flips heads 75 percent of the time. There are three coins. The tossing experiment always results in heads, and the message will always be 1. So, after 500 flips most of the probability gets distributed around the value 0. Here we will learn how to find the probability of tossing two coins. We use the experiement of tossing a coin three times to create the probability distribution table for the number of heads. (b) Solve the same problem for the case of blades of grass. I have a probability question. Similar Questions. (one coin flipped twice = two coins flipped at once, right?) Now, don't stop at 10, if you hit heads there, either, 11 heads in a row is 10 heads in a row twice, 12 is 3 times, et cetera. ) Suppose the probability that the two sides that land face up are the same color is 29 96 in the. There are three coins One is two headed coin, another is biased coin that comes up tails 25% of the times and the other is unbiased coin One of the three coins is chosen at random and tossed , it shows head What is the probability that - Math - Probability. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two heade. In my town, it's rainy one third of the days. What is the probability mass function of X? What is the expected value of X?. At the root (level 0) we choose randomly the first coin. (a) A gambler has in his pocket a fair coin and a two-headed coin. 3) The probability for two events to both occur, even if they are not independent, is the probability for the first to occur, times the probability for the second to occur given the condition that the first has already occurred. We begin with some previews. find probability that. 3 Bayes Formula From Prior to Posterior Bayesian Learning Examples: Manufacturing Bayes Continued, Bridged Circuit, Two-headed Coin. In the question as asked, you see only one side of the coin. Find the joint probability density function of (V, Y). In contrast, a process in which the. G is surprised to ﬁnd that he loses the ﬁrst ten times they play. What is the probability that the 2-headed coin is chosen? To solve this, note that Pf2-headed coing= 1 2; Pffair coing= 1 2: and Pfn headsj2-headed coing= 1; Pfn headsjfair coing= 2 n: By thelaw of total probability, Pfn headsg = Pfn headsj2-headed. This article shows you the steps for solving the most common types of basic questions on this subject. One is a two-headed coin (having a head on both faces), another is a biased coin that comes up heads 75% of the times, and third is also a biased coin that comes up tails 40% of the times. What’s the probability that it is the 2-headed coin? 9 Independence 30. A coin is selected at random and tossed. Problem 47: An urn contains 5 white and 10. Suppose that a bag contains 12 coins: 5 are fair, 4 are biased with probability of heads 1 3; and 3 are two-headed. A coin is chosen at random from the bag and tossed 2 times. So, if we ask the subject to guess heads or tails for each of 100 coin flips, we'd expect about 50 of the guesses to be correct. Since there are three coins, the probability that you chose the two-headed one is one-third. What is the probability that there is a head on the OTHER side of this coin? Yes, it could be the fair coin or the two-headed coin, but they're not equally likely: because the fair coin COULD have come up tails, the two-headed coin is now twice as likely. But what should the. Horace turns up at school either late or on time. Each unique arrangement (permutation) of possible coin tosses is equally likely. You believe it is equally likely that you picked either the 2-headed coin or the regular coin. With the coins it is the same thing: if I draw heads, it is more probable that the coin I picked is the double headed coin. There are three coins in a box. 1) and the probability you have a fair coin and it comes up. The correct reasoning is to calculate the conditional probability p= P(two-headed coin was chosen|heads came up) = P(two-headed coin was chosen and heads. Two of these coins are fair and the third coin is double-headed so that when tossed a head is always obtained. One is a two-headed coin, while the other two are normal. One coin is chosen at random and flipped, coming up heads. The correct reasoning is to calculate the conditional probability p= P(two-headed coin was chosenjheads came up) = P(two-headed coin was chosen and heads. EDIT #2: By "no retosses" I mean that your algorithm for obtaining the 1/3 probability can not have a "retoss until you get 1/3" rule which can theoretically cause you to toss infinitely many times. (a)A gambler has a fair coin and a two-headed coin in his pocket. When one coin is selected at random and ipped, it shows heads. Mint has made the following coins: A unique double-headed Jefferson nickel struck with 2 obverse dies. What is the probability that it is the fair coin? (b) Suppose that he ﬂips the same coin a second time and again it shows heads. One is two headed coin, another is biased coin that comes up tails 25% of the times and the other is unbiased coin. What is the. What are the possible outcomes? (List them) Answer: 2. What is the probability of throwing two heads in a row when tossing a coin? This is the same as asking what the probability that the first coin tossed will be head AND the second. Exercises for Chapter 1 Introduction to Probability Theory 1. What is the probability that you got the two-headed coin?. The ﬁrst is a two-headed coin, the second is a fair coin (so the chance of “heads” is 1/2), and the third is biased so that the chance of “heads” is 3/4. Assuming the coin is fair (has the same probability of heads and tails), the chance of guessing correctly is 50%, so you'd expect half the guesses to be correct and half to be wrong. We do not return it to the bin. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. One coin in a collection of 65 has two heads. It is ipped 100 times. Compute the proportion of. If heads appears both times, what is the probability that the coin is two-head. coin is chosen at random and flipped, and comes up heads. A coin is chosen at random from the bag and tossed. Simulate a random coin flip or coin toss to make those hard 50/50 decisions from your mobile Android, iPhone, or Blackberry phone or desktop web browser. heads on both sides. Alice tosses this coin repeatedly. I think hugin is right: the probability of a 6th head is just the combination of the probability you have the double-headed coin (0. For the sake of argument let's say 1 in 10,000 coins are double headed. You can & should always know. This is a basic introduction to a probability distribution table. You flip it and it comes up "heads". What is the probability it will come up heads the next time I flip it? “Fifty percent,” you say. tails—then we know it's not the two-headed coin. One coin has heads on both sides, one coin has tails on both sides, the third one has head on one side and tail on the other side. I think hugin is right: the probability of a 6th head is just the combination of the probability you have the double-headed coin (0. find probability that a:heads appear on the second toss. 1 Answer to (a) A gambler has a fair coin and a two-headed coin in his pocket. This is obviously an extreme example, but it illustrates that having a group of coins whose average probability of landing heads is 50% is not necessarily the. C onditional Independence Two events A and B are conditionally independent given a third event C precisely if the occurrence of A and the occurrence of B are independent events in their conditional probability. Exercises for Chapter 1 Introduction to Probability Theory 1. One is a two-headed coin, another is a fair coin, and the third is a biased coin which comes up heads 75% of the time. , HHH, HHT, HTH, HTT, THH, THT, TTH, TTT Out of which there are 4 set which contain at least 2 Heads i. Horace turns up at school either late or on time. The second one is a fair coin. A two-headed coin is a coin which has head on both sides; a fair coin means it has tail on one side and head on the other. So far we’ve established that: The probability of flipping heads or tails is equally likely each individual toss: P(H) = P(T) = 1/2. If the experiment can be repeated potentially inﬁnitely many times, then the probability of an event can be deﬁned through relative frequencies. if the face is heads what is the percentage chance that the other side is heads?. a) What is the probability that the coin chosen is the two-headed coin?. What is the probability that it is the fair coin? Hint: It is given that. Examples include a two-headed coin and rolling a die whose sides all show the same number. All k times the coin landed up heads. In the case of the coins, we understand that there's a $$\frac{1}{3}$$ chance we have a normal coin, and a $$\frac{2}{3}$$ chance it's a two-headed coin. The first coin is two-headed. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two head. Two-Face said that heads he would free Batman, tails he would kill, and Batman asked about the edge, and said he should agree to turn himself in and co-operate with all plastic surgery and. Suppose that there is a 99. Instead of probability distributions, we use probability densities and integrate over ranges of possible. Other than thisdierence, the coins are. There is a probability of 0. Probability must be between 0 and 1. What is the probability that she flipped the fair coin?. probability that it is a fair coin? (hint: for the fair coin, assumes the ﬂips are independent, the probability of getting two heads in two ﬂips equals 0. What is the probability that it was the two-headed coin? 8. They draw a coin from their pocket equally likely at random and ip the coin. The probability of A and B is 1/100. A gambler has in his pocket a fair coin and a two headed coin. For a two-headed coin, we might have P (X =H)=1 and P (X =T)=0. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. Case 2: One head. When one of the three coins is selected at random and ﬂipped, it shows heads. The outcome of the tosses comes up heads, heads, and heads. 5 of heads on each coin, so there is a 1/2 * 1/2 *1/2 = 1/8 = 0. A box contains three coins. What is the probability that it is the fair coin? Hint: It is given that. The third one is a biased coin that comes up heads $75\%$ of the time. More than 3 heads I don't know how to start that problem. Interview question for Quantitative Trader in Hong Kong. A box contains three coins: two regular coins and one fake, two{headed coin. If you want a discrete form of the question: There are 2 coins in a bag. We begin with some previews. The Coin Toss Probability Calculator an online tool which shows Coin Toss Probability for the given input. Coin flipping probability question They then install one, and see if it blows at the right set point. Every ip is a probability of some proposition H given some proposition E. What is the probability that you got the two-headed coin?. Heads, I Win There is a two-headed coin and fifteen fair coins. There are 3 coins in a box. Denote the event that the randomly selected coin lands heads up k times by B k. Let's split problem into two parts: 1) What is the probability you picked the double-headed coin (now referred as D)? 2) What is the probability of getting a head on the next toss?. Three ways to pick heads, two of them with the double headed coin, so the probability that the one in my hand has two heads is 2/3. You draw the two-headed coin and see face 2. No Fair, Two Heads Are Better Than One. What is the probability that she flipped the fair coin?. 5 Random numbers and conditional execution: flipping a coin. chance of choosing one of the other coins, and getting two heads - 4/5*1/2*1/2 or 20% so there is a 40% chance of getting two heads in a row with any randomly chosen coin, so the probability that the 2 headed coin was chosen should be 20/40 or 50% so there was a 50% chance that it was the two-headed coin. What is the probability that they get the same number of tails? he does NOT use a two-headed coin, then on! question "the. a coin is selected at random and tossed. The answer is 1/2. You get a likelihood ratio of, of eight and in this case that means that there's eight times as much evidence supporting the hypothesis that the coin is two headed relative to the hypothesis that the coin is fair. If the first coin flipped is a T, then the remaining flips must fall under one of the configurations of. Associated Topics || Dr. C onditional Independence Two events A and B are conditionally independent given a third event C precisely if the occurrence of A and the occurrence of B are independent events in their conditional probability. Problem 16 Solution Yes, the answer is 2/3. ) P (coin showing heads. ip(HjE) = 1=2; ip(HjE&E0) = 1. 1 so, overall probability that the next flip is heads =. What is the probability that two headed coin was selected? Denote with Ak the event that randomly selected coin lands heads up k times. Given that you see 10 heads, what is the probability that the next toss of that coin is also a head?. What is the probability that the coin chosen is a two headed coin. 75 since three time out of four a side will be heads: 3) The probability is 2/3 since the double headed coin has twice the chance of showing a head face up. The rest are fair. " A coin is selected at random, ﬂipped n times and in all ﬂips it falls heads up. Classical probability says that a fair coin has a 50-50 chance of coming up heads or tails. Thus, we have the conditional probability $P(F \mid E_1) = 1$. There are 3 coins in a box. The correct reasoning is to calculate the conditional probability p= P(two-headed coin was chosenjheads came up) = P(two-headed coin was chosen and heads. 2- There are three coins in a box. Then, given you get head after tossing, then chance that you chose double head coin is (1/3)/(1/3+1/6) = 2/3. chance of choosing one of the other coins, and getting two heads - 4/5*1/2*1/2 or 20% so there is a 40% chance of getting two heads in a row with any randomly chosen coin, so the probability that the 2 headed coin was chosen should be 20/40 or 50% so there was a 50% chance that it was the two-headed coin. This is, however, wrong, because given that heads came up, it is more likely that the two-headed coin was chosen. One of the three coins is chosen at random and tossed, and it shows heads. Chapter 3 Probability contrast, if we flip a two-headed coin we do know for sure what the result will be, heads! An experiment in which any one of number of possible outcomes may result is called a random experiment or probability experiment.   Hence, 2/3. You are given a coin to test. One sold for $41,975 while another took$75,000. What is the probability that it is the fair coin? (b) Suppose that he flips the same coin a second time and, again, it shows heads. F • Fair coin - a fair coin is defined as coin where the probability of landing heads up or tails up are the same (0. a) Find the probability that the first two marbles selected are black and the next two white. Suppose we have 3 unbiased coins and we have to find the probability of getting at least 2 heads, so there are 2 3 = 8 ways to toss these coins, i. What is the conditional probability that. It comes up heads each time. Now let's flip a coin twice in succession. This is, however, wrong, because given that heads came up, it is more likely that the two-headed coin was chosen. Solution 2. A box has three coins. One is a two-headed coin; another is a fair coin; and the third is a biased coin that comes up heads 75% of the time. If G's prior belief is that the chance of R having a two headed coin is 0. then another coin is selected from the two remaining coins and tossed. Depending on which coin you have, there is a 50% chance that the other side is tails (regular coin) and a 50% chance that the other side is heads (two-headed coin). Prisoner A asks the jailer to tell him. (a) A gambler has a fair coin and a two-headed coin in his pocket. The second one is a fair coin. One sold for $41,975 while another took$75,000. Conditional probability. the opposite face is either heads or tails, the desired probability is 1/2. A couple of two-tailed Washington quarters were also made by the U. One coin is chosen at random and tossed twice. Three ways to pick heads, two of them with the double headed coin, so the probability that the one in my hand has two heads is 2/3. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. I think hugin is right: the probability of a 6th head is just the combination of the probability you have the double-headed coin (0. Given that heads comes up, what is the probability that you flipped the two headed coin?. Problem 43: There are 3 coins in a box. Suppose that he ﬂips the same coin a third time and it shows tails. Let's split problem into two parts: 1) What is the probability you picked the double-headed coin (now referred as D)? 2) What is the probability of getting a head on the next toss?. Probability of heads coming up, given that it is a biased coin= 75% Since the third coin is unbiased, the probability that it shows heads is always. What is the probability that it is the fair coin? (b) Suppose that he ﬂips the same coin a second time and again it shows heads. In the case of the coins, we understand that there's a $$\frac{1}{3}$$ chance we have a normal coin, and a $$\frac{2}{3}$$ chance it's a two-headed coin. One coin is chosen at random and tossed twice. I'm not a mathematician so please bear with me. then another coin is selected from the two remaining coins and tossed. chance of choosing one of the other coins, and getting two heads - 4/5*1/2*1/2 or 20% so there is a 40% chance of getting two heads in a row with any randomly chosen coin, so the probability that the 2 headed coin was chosen should be 20/40 or 50% so there was a 50% chance that it was the two-headed coin. Example: H = the coin lands heads, E = the coin is either two-headed or two-tailed and is about to be tossed, E0 = the coin has a head on one side. You draw the two-headed coin and see face 2. There are 3 coins, 2 fair and 1 is a two headed coin. A two-headed coin will always show heads.   - 1843971. One has two heads, another two tails and the last is a fair coin. What is the probability that it is the fair coin?. the coin ip, they are now dependent: if you were to go on to discover that the coin has two heads, the hypothesis of psychic powers would return to its baseline probability { the evidence for psychic powers was \explained away" by the presence of the two-headed coin. The possible outcomes are heads or tails. Explanation: Assume each coin is chosen with probability 1 2 and consider a single ip. There are three coins in a box. What is the probability that the two-headed coin was selected? Vidakovic (GaTech) (R)Evolution in Statistics May 19, 2015 5 / 35. If heads appears both times, what is the probability that the coin is two-head. Now what is the probability that it is the fair coin?. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. It's more complicated than other games. Interview question for Quantitative Trader in Hong Kong. the opposite face is either heads or tails, the desired probability is 1/2. This is, however, wrong, because given that heads came, it is more likely that the two-headed coin was chosen. There are 3 coins in a box. We already know that one side is heads, the question is what is the chance that the other side is tails. So P(X=1) = P(choose a 1 headed coin) x P(1 head, 1 tail obtained) = 4/6 x 1/2 x 1/2 = 1/6 If X = 2, there are 2 scenarios:. This is the only known U. Two coins are available, one fair and the other two-headed. The correct reasoning is to calculate the conditional probability p = P(two-headed coin was chosen|heads came) = P(two-headed coin was chosen and heads came). unbiased coin chosen, so unlike in (a), we can get a white and a blue face for the two coins ﬂipped. What is the probability that it is the fair coin? 2. What is the probability that it is the fair coin? (b) Suppose that he flips the same coin a second time and, again, it shows heads. c) Calculate the probability of red or green on the spinner and tail on the coin.   - 1843971. If it is heads, he is willing. 1) and the probability you have a fair coin and it comes up. a) What is the probability that the coin chosen is the two-headed coin?. One is a two headed coin (having head on both faces),another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. For continuous random variables, there is a subtle problem. Simulate a random coin flip or coin toss to make those hard 50/50 decisions from your mobile Android, iPhone, or Blackberry phone or desktop web browser. The hypotheses areH1-the coin is two headed, and H2 the coin is fair. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin? (b) Suppose that he ﬂips the same coin a second time and again it shows heads. Remember that P(A given B) = P(A and B)/P(B) So let's say that A is the event that he chose the 2-headed coin, and B is an event denoted by H(N), which indicates that the coin was tossed N times, and came up heads each time, so the answer in our first case is P(A given H(1)), and the answer our last case is P(A given H(3)). (one coin flipped twice = two coins flipped at once, right?) Now, don't stop at 10, if you hit heads there, either, 11 heads in a row is 10 heads in a row twice, 12 is 3 times, et cetera. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. (b) two of clubs? Example 4.