A hockey player is to take 3 shots on a certain goalie. The probability he will

Question

A hockey player is to take 3 shots on a certain goalie. The probability he will score a goal on his first shot is 0.55. If he scores on his first shot, the chance he will score on his second shot increases by 0.1; if he misses, the chance that he scores on his second shot decreases by 0.1. This pattern continues to on his third shot: If the player scores on his second shot, the probability he will score on his third shot increases by another 0.1; should he not score on his second shot, the probability of scoring on the third shot decreases by another 0.1. A random variable X X counts the number of goals this hockey player scores. (a) Finish the probability distribution of X which is a table that looks like

X = 0. 1. 2. 3. 4.

P(x)= ? ? ? ? ?

Explanation / Answer

here below is pmf of X:

P(X=0)=P(all three shorts are failure) =0.45*0.55*0.65=0.160875

P(X=1) =P(hit on first shot and fail on next 2+fail on first hit on 2nd and fail on 3rd+fail on first 2 and hit on 3rd)

=0.55*0.35*0.45+0.45*0.45*0.45+0.45*0.55*0.35=0.264375

P(X=2)=P(hit on first 2 and fail on 3rd+hit on first fail nd and hit on 3rd+fail on first and hit on last 2)

=0.55*0.65*0.25+0.55*0.35*0.55+0.45*0.45*0.55=0.306625

P(X=3) =P(hit on all three) =0.55*0.65*0.75=0.268125

hence pmf of X:

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