Question

7.- A balanced coin is thrown until heads or 3 tails appears, whichever comes first. Let X be the number of releases required. Obtain the probability function of the random variable X to calculate the expected number of releases. Round the result to two decimal places.

Answer 1

7.

The possible Scenarios and the number of releases required are as follows

H : Head : Number of releases : X=1

TH : Tail, Head : Number of releases : X=2

TTH : Tail, Tail, Head : Number of releases : X=3

TTT: Tail, Tail, Tail : Number or releases : X=3

P(X=1) = P(H) = 1/2 =0.5

P(X=2) = P(TH)=P(T)P(H) = (1/2)*(1/2) = (1/4)=0.25

P(X=3) = P(TTH)=P(T)P(T)P(H) = (1/2)*(1/2)*(1/2) = (1/8)=0.125

P(X=3) = P(TTT)=P(T)P(T)P(T) = (1/2)*(1/2)*(1/2) = (1/8)=0.125

P(X=3) = P(TTH) +  P(TTT) = 0.125+0.125 = 0.25

Therefore Probability function of X is

X	P(X)
1	0.5
2	0.25
3	0.25

expected number of releases = E(X)

expected number of releases = 1.75