Each day John performs the following experiment. He flips a fair coin repeatedly until he sees...

Each day John performs the following experiment. He flips a fair coin repeatedly until he sees a T and counts the number of coin flips needed.

(a) Approximate the probability that in a year there are at least 3 days when he needed more than 10 coin flips. Argue why this approximation is appropriate.

(b) Approximate the probability that in a year there are more than 50 days when he needed exactly 3 coin flips. Argue why this approximation is appropriate.

Expert Solution

with continuity correction in (b), the probability is .26984.

For query in above, comment.

orchestra answered 1 year ago

A fair coin is flipped until a head appears. Let the number of flips required be...

A fair coin is flipped until a head appears. Let the number of flips required be denoted N (the head appears on the ,\1th flip). Assu1ne the flips are independent. Let the o utcon1es be denoted by k fork= 1,2,3, . ... The event {N = k} 1neans exactly k flips are required. The event {,v;;, k} n1eans at least k flips are required. a. How n1any o utcon1es are there? b. What is Pr[N = k] (i.e., the probability...

An experiment consists of repeatedly tossing 2 fair coins until the toss results in one each...

An experiment consists of repeatedly tossing 2 fair coins until the toss results in one each of a Head and a Tail. What is the mathematical expectation of the number of times you will need to toss the 3 coins to achieve this?

A fair coin is tossed repeatedly until it has landed Heads at least once and has...

A fair coin is tossed repeatedly until it has landed Heads at least once and has landed Tails at least once. Find the expected number of tosses.

Consider a series of 8 flips of a fair coin. Plot the entropy of each of...

Consider a series of 8 flips of a fair coin. Plot the entropy of each of the 8 possible outcomes. Entropy is in J/K×1023J/K×1023 Entropy Number of heads012345678012345678910

A fair coin is flipped six times. The outcomes of the coin flips form a palindrome...

A fair coin is flipped six times. The outcomes of the coin flips form a palindrome if the sequence of T’s and H’s reads the same forwards and backwards, e.g. THTTHT. Let A denote the event that the first, second and fourth flips are all ‘T’. Let Z denote the event that the six flips form a palindrome. (a) Is A independent of Z? (b) Is A independent of Z? (c) A fair coin flipped six times and a certain...

A fair coin is tossed until the first head occurs. Do this experiment T = 10;...

A fair coin is tossed until the first head occurs. Do this experiment T = 10; 100; 1,000; 10,000 times in R, and plot the relative frequencies of this occurring at the ith toss, for suitable values of i. Compare this plot to the pmf that should govern such an experiment. Show that they converge as T increases. What is the expected number of tosses required? For each value of T, what is the sample average of the number of...

There is a fair coin and a biased coin that flips heads with probability 1/4.You randomly...

There is a fair coin and a biased coin that flips heads with probability 1/4.You randomly pick one of the coins and flip it until you get a heads. Let X be the number of flips you need. Compute E[X] and Var[X]

Q. (a) Tegan is trying to decide if a coin is fair. She flips it 100...

Q. (a) Tegan is trying to decide if a coin is fair. She flips it 100 times and gets 63 heads. Please explain why it might make sense to view 63 heads as enough evidence to conclude the coin is unfair. (b) Leela is trying to decide if a coin is fair. She flips it 100 times and gets 63 heads. Please explain why it might NOT make sense to view 63 heads as enough evidence to conclude the coin...

In a sequence of independent flips of a fair coin, let N denote the number of...

In a sequence of independent flips of a fair coin, let N denote the number of flips until there is a run of three consecutive heads. Find P(N ≤ 8). (Should write out transition matrix.)

Given that in 4 flips of a fair coin there are at least two "heads", what...

Given that in 4 flips of a fair coin there are at least two "heads", what is the probability that there are two "tails"? There are ten equally likely outcomes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. You randomly select one value, call it the initial value. Then, you continue to randomly select values, call them follow-up selections, until you come up with the initial value. What is the fewest number of follow-up selections that insures that...

Question