Question

A fair coin is tossed. If the toss results in a head, then one die is thrown, while if the toss results in a tail, then two dice are thrown. Let X denote the random variable that counts the number of spots showing on the thrown die or dice. The values that X can assume are the positive integers from 1 to 12 inclusive. Find the following probabilities. Your answers should be whole numbers or fractions in lowest terms. Pr(X=1) Pr(X=5) Pr(X=10) Pr(X<10 )

Answer 1

P(X=1)=P(heads comes and 1 die shows 1) =(1/2)*(1/6)=1/12

P(X=5)=P(heads comes and 1 die shows 5) +P(tails comes and 2 die shows sum as 5)=(1/2)*(1/6)+(1/2)*(4/36)

=10/72=5/36

P(X=10)=P(tails comes and 2 die shows sum as 10)=(1/2)*(3/36)=1/24

P(X<10)=1-(P(X=10)+P(X=11)+P(X=12))=1-P(tails comes)*(P(10 comes as sum+11 comes as sum+12 comes as sum))=1-(1/2)*(3/36+2/36+1/36)=1-1/12=11/12