• Aug 10th 2008, 10:30 AM
chopet
A game requires you to shoot basketballs with the balls landing in either of the 3 baskets labelled A,B and C. Your ball lands in A with the probability of 1/2, and you get paid $1. It lands in B with the probability of 1/4, and you get paid$2, and it lands in C with the probability of 1/4, and you get paid $0. Assuming that you are given 50 balls to shoot for free, what is the probability that you will earn at least$30?
• Aug 11th 2008, 10:14 AM
TKHunny
50 balls! This is a nicely large sample to suggest the use of the Normal Distribution as a reasonable approximation to whatever the given distribution is.

Distribution for 1 ball

X = Payoff Random Variable

E[X] = (1/2)(1) + (1/4)*(2) + (1/4)(0) = 1

Mean(X) = 1

E[X^2] = (1/2)(1)^2 + (1/4)*(2)^2 + (1/4)(0)^2 = 1.5

Var(X) = E[X^2] - (E[X])^2 = 1.50 - 1 = 0.5

Then we have:

Mean(50*X) = 50*Mean(X) = 50
Var(50*X) = (50^2)*Var(X) = 1250
Standard Deviation (50*X) = sqrt(Var(50*X)) = 35.355

Now what?
• Aug 12th 2008, 06:32 AM
CaptainBlack
Quote:

Originally Posted by TKHunny
50 balls! This is a nicely large sample to suggest the use of the Normal Distribution as a reasonable approximation to whatever the given distribution is.

Distribution for 1 ball

X = Payoff Random Variable

E[X] = (1/2)(1) + (1/4)*(2) + (1/4)(0) = 1

Mean(X) = 1

E[X^2] = (1/2)(1)^2 + (1/4)*(2)^2 + (1/4)(0)^2 = 1.5

Var(X) = E[X^2] - (E[X])^2 = 1.50 - 1 = 0.5

Then we have:

Mean(50*X) = 50*Mean(X) = 50
Var(50*X) = (50^2)*Var(X) = 1250
Standard Deviation (50*X) = sqrt(Var(50*X)) = 35.355

Now what?

The pay out on each ball is a RV independent of the payout on any other ball but with identical distribution.

The variance of the sum on $N$ iid RV of variance $\sigma^2$ is $N\sigma^2$.

(The probability of a payout of less than $30 is negligable, and so far in the wings of the distribution that a normallity assumption would not be valid for anything other than suggesting the probability is "small") RonL • Aug 14th 2008, 05:14 PM Nobby program a solution I think one way to find an answer is to write a simple piece of code and run 100000 examples and count how many times you get less than$30.

You could also write a piece of code to calculate the actual value. To calculate an exact answer by hand would be very time consuming.

just my two penneth...

If you are not sure how to code in something like Excel, you should try to learn, it is a great skill to have.

cheers

Nobby.