Question

John is going to buy baseball cards until he gets his favorite player. John only has enough money to buy 5 packs of cards. Suppose that each pack has a 1 out of 5 chance of containing the card John is looking for Let the random variable X be the number of packs of cards John buys. Compute the probability mass function for X, and compute P(X2 3).

Answer 1

below is pmf of X:

P(X=1)=P(first deck contains the card John is looking for) =1/5

P(X=2)=P(first deck not but second deck contain the card) =(4/5)*(1/5)=4/25

P(X=3)=P(first 2 deck not contains but third pack contains)=(4/5)²*(1/5)=16/125

P(X=4)=P(first 2 deck not contains but third pack contains)=(4/5)³*(1/5)=64/625

P(X=5)=P(first 4 pack not contains)=(4/5)⁴ =256/625

hence P(X>=3) =P(X=3)+P(X=4)+P(X=5)=(16/125)+(64/625)+(256/625)=400/625=16/25