You toss a penny and observe whether it lands heads up or tails up. Suppose the...
1. Consider flipping a fair coin three times and observe whether it lands heads up or tails up. Let X the number of switches from either head to tail or vice versa. For example, when THT is observed, the number of switches is 2 and when HHH is observed, the number of switches is 0. Also, let Y be the number of tails shown in the three times of fipping. (a) List all the values of the joint probability mass...
You have a biased coin where heads come up with probability 2/3 and tails come up with probability 1/3. 2. Assume that you flip the coin until you get three heads or one tail. (a) Draw the possibility tree. (b) What is the average number of flips? Use the possibility tree, and show your calculation. 2. Assume that you flip the coin until you get three heads or one tail. (a) Draw the possibility tree. (b) What is the average...
Suppose that I toss a fair coin 100 times. Write 'p-hat' for the proportion of Heads in the 100 tosses. What is the approximate probability that p-hat is greater than 0.6? 0.460 0.023 0.540 We can't do the problem because we don't know the probability that the coin lands Heads uppermost 0.977
Q3. Suppose we toss a coin until we see a heads, and let X be the number of tosses. Recall that this is what we called the geometric distribution. Assume that it is a fair coin (equal probability of heads and tails). What is the p.m.f. of X? (I.e., for an integer i, what is P(X=i)? What is ?[X]? ({} this is a discrete variable that takes infinitely many values.)
Suppose you flip an ordinary fair coin 60 times and amazingly it lands on heads every single time. What is the probability that on your next flip, it lands on tails?
We are given three coins. One has heads on both faces, the second has tails on both faces, and the third coin has a head on one face and a tail on the other face. We choose one coin at random, toss it, and observe that the result is heads. What is the probability that the opposite face is tails?
3. We are given three coins. One has heads on both faces, the second has tails on both faces, and the third coin has a head on one face and a tail on the other face. We choose one coin at random, toss it, and observe that the result is heads. What is the probability that the opposite face is tails?
Suppose we flip a fair coin n times. We say that the sequence is balanced when there are equal number of heads and tails. For example, if we flip the coin 10 times and the results are HT HHT HT T HH, then this sequence balanced 2 times, i.e. at position 2 and position 8 (after the second and eighth flips). In terms of n, what is the expected number of times the sequence is balanced within n flips?
If we repeatedly toss a balanced coin, then, in the long run, it will come up heads about half the time. But what is the probability that such a coin will come up heads exactly half the time in 26 tosses?
Problem 2 Suppose you flip a penny and a dime. Each coin is equally likely to come up heads and tails. The two flips are independent a) What is the sample space? b) What is the conditional probability that both coins come up heads, given that the penny comes up heads? c) What is the conditional probability that both coins come up heads, given that at least one of the coins comes up heads? (Hint: the answers in part (b)...