I need full explanation of how to do this question. Please answer the two parts;

Question

I need full explanation of how to do this question. Please answer the two parts; otherwise, don’t attempt it.
In a carnival sharpshooter game, the shooter pays $10 and takes successive shots at a target until he misses. Each time he hits the target, he gets back $3. The game is over as soon as he misses a target. The sharpshooter estimates his probability of hitting the target on any given shot as p. According to his estimate, he expects to gain $2 on the game.
a) Find p. b) Find the probability that the shooter gains at least $8. I need full explanation of how to do this question. Please answer the two parts; otherwise, don’t attempt it.
In a carnival sharpshooter game, the shooter pays $10 and takes successive shots at a target until he misses. Each time he hits the target, he gets back $3. The game is over as soon as he misses a target. The sharpshooter estimates his probability of hitting the target on any given shot as p. According to his estimate, he expects to gain $2 on the game.
a) Find p. b) Find the probability that the shooter gains at least $8. I need full explanation of how to do this question. Please answer the two parts; otherwise, don’t attempt it.

a) Find p. b) Find the probability that the shooter gains at least $8.

Explanation / Answer

given

shooter paid =$10

he gets $3 for each hit

P(hitting the target)=p

so P(fails to hit the target)=1-p=q

let X is number of times shooter hits the target before he missed

so X is Geometric distribution with Probability =1-p=q

so

P(X=x) =pxq

a)

he estimates gain of $2 that means he estimates getting $12

so he estimates that he will hit 4 times target before getting missed

so

E(X) =4

as we know that estimate of Negative bionomial is given by (failure prob)/(succes prob)

here our case succes prob =1-p and failure prob =p

hence

p/(1-p) =4

hence

p=4-4p

this gives p=4/5 =0.8

b)

we have to find probability that shooter gains at least $8

so for gain of $8 ,shooter needs to get $18 by hitting the target means at least 6 times he will hit the target

so

we have to find P(X>6)=?

Now

P(X>6)=1-P(X<6) =1-[P(X=5) +P(X=4)+P(X=3)+P(X=2)+P(X=1)+P(X=0)]

=1-[ (0.8)5*0.2 +(0.8)4*0.2 +(0.8)3*0.2+(0.8)2*0.2 +0.8*0.2+0.2]

=1-[ 0.065 +0.082 +0.102 +0.128 +0.16 +0.2]

=1- 0.737 =0.263

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