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 of a sequence of k - 1 tails followed by a heads)? (Hint: write a gene ral expression for Pr[N = k] for any k = 1,2,3, .. . )

c. Show the probabilities sum to l (i.e., I:f: 1 Pr[,v = k] = 1).

d. What is Pr [ N ;;, I] for all I;;: l?

e. What is Pr[N s /] for all I;;: l?

f. Do the answers to tl1e previous two parts sum to l? Should they?

Expert Solution

Geometric distribution

milcah answered 1 year ago

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 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...

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.)

The second game You toss a fair coin until Head appears and you are paid the...

The second game You toss a fair coin until Head appears and you are paid the number of Tails received, for example, if we obtain TTTH you get 3 rubles. How much do you agree to pay the host for the game? The third and fourth games We have a deck of 52 cards. a) We randomly choose 5 cards (without repetition) and get 5 rubles for any ace or king chosen; b) We randomly choose 5 cards (with repetition,...

Let discrete random variable X be the number of flips of a biased coin required to...

Let discrete random variable X be the number of flips of a biased coin required to get tails, where P(tails) = 1/3 . a) Calculate the probability for every value of X from 1 to 10. b) Sketch a plot of the p.m.f. of X for the first 10 flips. c) Sketch a plot the c.d.f. of X for the first 10 flips.

An experiment is to flip a coin until a head appears for the first time. Assume...

An experiment is to flip a coin until a head appears for the first time. Assume the coin may be biased, i.e., assume that the probability the coin turns up heads on a flip is a constant p (0 < p < 1). Let X be the random variable that counts the number of flips needed to see the first head. (a) Let k ≥ 1 be an integer. Compute the probability mass function (pmf) p(k) = P(X = k)....

You roll two fair four-sided dies and then flip a fair coin. The number of flips...

You roll two fair four-sided dies and then flip a fair coin. The number of flips is the total of the roll. a. Find the expected value of the number of heads observed. b. Find the variance of the number of heads observed.

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)...

8. A fair coin is tossed 60 times. Find the probability that the head appears between...

8. A fair coin is tossed 60 times. Find the probability that the head appears between 22 and 40 times by using a. binomial distribution, b. approximation of Binomial distribution by normal distribution. Discuss why b. is better in practice.

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...

Question