prags1989 wrote:

I still don't understand this question. Can someone help?

If the probability that an unfair coin displays heads after it’s flipped is three times the probability that it will display tails, what is the probability that, after 5 coin flips, the coin will have displayed heads exactly 3 times?A. 27/1024

B. 135/512

C. 9/64

D. 5/32

E. 11/32

A fair coin has the probability of heads equal to the probability of tails, so P(tail) = P(head) = 1/2. We are told that the coins is unfair so P(tail) ≠ P(head).

The probability that the coin displays heads after it’s flipped is three times the probability that it will display tails: P(h) = 3*P(t). Sin, the sum of these probabilities must be 1, then P(h) + P(t) = 1:

3*P(t) + P(t) = 1;

P(t) = 1/4 and P(h) = 3/4.

The question ask to find the probability that, after 5 coin flips, the coin will have displayed heads exactly 3 times. Exactly 3 heads can occur in several different ways:

HHHTT

HHTHT

HTHHT

THHHT

THHTH

THTHH

TTHHH

HHTTH

HTTHH

HTHTH

Basically, this is permutations of 5 letters HHHTT, out of which 3 H's and 2 T's are identical: 5!/(3!2!) = 10.

Now, each of the above 10 cases have the probability of (3/4)^3*(1/4)^2. Thus, the overall probability of P(HHHTT) is \(10*(\frac{3}{4})^3*(\frac{1}{4})^2 =\frac{135}{512}\).

Answer: B.

Hope it's clear.