Question

In: Math

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.

Solutions

Expert Solution

Let be the probability of landing Heads.

The experiments will terminate in tosses,

1) When the toss is a Head preceded by Tails. Whose probability is

2) When the toss is a Tail preceded by Heads. Whose probability is

The above 2 events are disjoint.

The probability that the experiments will terminate in tosses is

Here . Hence,

The expected tosses is

Note  that is the expected value of a Geometric distribution which is 2. Thus,


Related Solutions

A coin is tossed repeatedly until heads has occurred twice or tails has occurred twice, whichever...
A coin is tossed repeatedly until heads has occurred twice or tails has occurred twice, whichever comes first. Let X be the number of times the coin is tossed. Find: a. E(X). b. Var(X).
A coin has a probability of 1/4 for head, and is repeatedly tossed until we throw...
A coin has a probability of 1/4 for head, and is repeatedly tossed until we throw head. The successive results of the toss are independent of each other. What is the probability that the first time we throw head after an odd number of toss? Hint: Use the law of total probability and consider the event that the first toss is head is, and her complement, as conditioning events. Correct answer: 3/7
A coin is tossed 400 times, landing heads up 219 times. Is the coin fair?
A coin is tossed 400 times, landing heads up 219 times. Is the coin fair?
If a fair coin is tossed 25 times, the probability distribution for the number of heads,...
If a fair coin is tossed 25 times, the probability distribution for the number of heads, X, is given below. Find the mean and the standard deviation of the probability distribution using Excel Enter the mean and round the standard deviation to two decimal places. x   P(x) 0   0 1   0 2   0 3   0.0001 4   0.0004 5   0.0016 6   0.0053 7   0.0143 8   0.0322 9   0.0609 10   0.0974 11   0.1328 12   0.155 13   0.155 14   0.1328 15   0.0974 16  ...
A fair coin is tossed r times. Let Y be the number of heads in these...
A fair coin is tossed r times. Let Y be the number of heads in these r tosses. Assuming Y=y, we generate a Poisson random variable X with mean y. Find the variance of X. (Answer should be based on r).
A coin with P[Heads]= p and P[Tails]= 1p is tossed repeatedly (the tosses are independent). Define...
A coin with P[Heads]= p and P[Tails]= 1p is tossed repeatedly (the tosses are independent). Define (X = number of the toss on which the first H appears, Y = number of the toss on which the second H appears. Clearly 1X<Y. (i) Are X and Y independent? Why or why not? (ii) What is the probability distribution of X? (iii) Find the probability distribution of Y . (iv) Let Z = Y X. Find the joint probability mass function
A coin is flipped repeatedly until either two heads appear in a row or two tails...
A coin is flipped repeatedly until either two heads appear in a row or two tails appear in a row(and then stop). Find the exact answer for P(two heads in a row appears before two tails in a row) for a coin with probability p of getting heads.
Toss a fair coin repeatedly. Let N1 be the number of tosses required to obtain heads...
Toss a fair coin repeatedly. Let N1 be the number of tosses required to obtain heads followed immediately by tails. Let N2 be the number of tosses required to obtain two heads in a row. (A) Should N1 and N2 have the same expected value? If not, which expected value should be larger? Explain your answers. (B) Find the probability mass function of N1. (C) Find the expected value of N1. (D) Find the probability mass function of N2. (E)...
A fair coin is tossed until a head appears. Given that the first head appeared on...
A fair coin is tossed until a head appears. Given that the first head appeared on an even-numbered toss, find the probability that it occurred on the second or the fourth toss.
A fair coin is tossed four times. Let X denote the number of heads occurring and...
A fair coin is tossed four times. Let X denote the number of heads occurring and let Y denote the longest string of heads occurring. (i) determine the joint distribution of X and Y (ii) Find Cov(X,Y) and ρ(X,Y).
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT