# At the Statesville County Fair, the probability of winning a prize in the basketball toss game is 0.1. Show the probability distribution for the number of prizes won in 8 games. If the game will be played 500 times during the fair, how many prizes should the game operators keep in stock? (a) We are given that the probability of winning p=0.1, and we are asked to give the probability distribution for n=8 trials.

This is an example of a binomial distribution: there are two outcomes (win/lose), the outcomes are independent, the probability does not change, and there are a finite number...

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(a) We are given that the probability of winning p=0.1, and we are asked to give the probability distribution for n=8 trials.

This is an example of a binomial distribution: there are two outcomes (win/lose), the outcomes are independent, the probability does not change, and there are a finite number of trials. Thus we can use the binomial probability to compute the individual probabilities.

A probability distribution consists of a number of possible events and their associated probabilities.

P(0 wins)= `(,)(.1)^0(.9)^8~~.4305 ` where the first factor is the number of combinations of 8 items choosing 0.

P(1 win)= `(,(.1)^1(.9)^7~~.3826 `

etc... (These could also be computed with a statistical package in a graphing calculator or with something like Excel.)

The probability distribution:

X    P(X)
0    .4305
1    .3826
2    .1488
3    .0331
4    .0046
5    .0004
6    `2.3"x"10^(-5) `
7    `7.2"x"10^(-7) `
8    `1"x"10^(-8) `   or `(.1)^8 `

You can display this graphically using the probabilities along the vertical axis and the events along the horizontal axis.

(b) If the game is played 500 times, we can use the expected value to compute the number of prizes to keep on hand. The expected value is given by `mu=np ` , so here we have 500(.1)=50.

(In the real world, you would want to keep a few more than 50.)

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