1. A coin is flipped four times. Assume all outcomes are equally likely. What is the probability that there are at least two consecutive flips that come up heads? 2. A hand of 5 cards is drawn from a standard deck of 52 cards. Assume all outcomes are equally likely. What is the probability that at least two of the cards have the same rank? Hint: Find the probability of the complement.
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- cs hw helpA fair coin is flipped 12 times in a row. a What is the probability that it lands on tails exactly one third of the times? b What is the probability that it lands on heads at least twice? c What is the probability that the first flip is heads or the last flip is heads?9. Prove the formula: e* ≥ 1 + x, Vx € R. 10. What is the probability that in a random group of 10 people at least two have the same birthday? Consider the experiment: Two fair die are cast and the sum of the spots shown on the uppermost
- Problem 1. You are playing a version of the roulette game, where the pockets are from 0 to 10and even numbers are red and odd numbers are black (0 is green). You spin 3 times and add up the values you see. What is the probability th at you get a total of 17 given on the first spin you spin a 2? What about a 3? Solve by simulation and analytically.A detective has interviewed four witnesses to a crime. From the stories of the witnesses the detective has concluded that: If the handyman is lying, then so is the butler. The gardener and the handyman cannot both be telling the truth. The cook and the gardener are not both lying. The handyman is telling the truth only if the cook is lying. For each of the four witnesses, can the detective determine whether that person is telling the truth or lying? Use the truth table to explain your reasoning (Give the full truth table). Please use the following notations in your reasoning – B: The butler is telling the truth. C: The cooking is telling the truth. G: The gardener is telling the truth. H: The handyman is telling the truth.A box contains three fair coins, and two trick coins that have heads on both sides.(a) What is the probability that a randomly selected coin from the box will come up heads whenflipped? (b) Suppose you randomly pick a coin from the box and flip it, and it comes up heads. What is theprobability that it’s one of the trick coins? (c) Let X be the random variable that represents the number of heads when every coin in the box isflipped. What is the probability distribution of X? (Hint: you can try counting all the outcomes,or you can remember that coin flips are independent of each other.) (d) What is the expected number of heads when all the coins are flipped?Page 2
- Assume that, with probability 0.8, It’s going to snow and/or I’m going to stay at home; and with probability 0.7, when it snows, then I stay at home. What is the probability of being truth that I am going to stay at home?6. AND Since there are two zeros on the board, let's determine the probability that you would be able to cover them on 2 consecutive turns. This means you'd have to roll a zero and then roll another zero. You can think of this as an AND probability. You need to roll 0 AND then you need to roll 0 again. Since the result of the first roll will have no influence on the result of the second roll, you can use the following rule: P(A and B) = P(A) * P(B) a) Determine P(0 and 0). b) Determine 1 - P(0 and 0). c) Write a sentence describing the meaning of 1 - P( Oand 0 ).Recall the airplane puzzle from the lab: the first person to get on the plane picks a random seat, the rest sit in their assigned seat if it is available, otherwise pick a random available seat. Assume person i is assigned the seat i. Suppose person 18 came in and discovered that their seat is taken. Assuming that the last person's seat, which is seat number n=32 on this plane, it is still available, what is the probability that person 18 will randomly pick it? Hint: to make sure your formula is correct, check that it works for the passenger right before the last one. Enter your answer as a decimal number with at most three digits after the decimal point, rounding if necessary. For example, if your answer is 1/3, round it and enter it as 0.333, if your answer is 2/3, enter it as 0.667, and if your answer is 1/2, enter it as 0.5. Your Answer: