Question

a gambler is going to play a gambling game. in each game, the chance of winning $3 is 2/10, the chance of losing $2 is 3/10, and the chance of losing $1 is 5/10. suppose the gambler is going to play the game 5 times. (a) write down the box model for keeping track of the net gain and the box model for keeping track of the number of winning plays. (b) calculate the expected value and standard error for the number of winning plays. (c) would it be appropriate to use the normal approximation for the number of winning plays? why or why not?

Answer 1

(a) Box model for keeping track of net gain:

Win $3 with probability 2/10 (represented by +3)

Lose $2 with probability 3/10 (represented by -2)

Lose $1 with probability 5/10 (represented by -1)

Box model for keeping track of the number of winning plays:

Win with probability 2/10

Lose with probability 8/10

(b) The expected value for the number of winning plays can be calculated as:

E(X) = np = 5 * 2/10 = 1

The variance can be calculated as:

Var(X) = np(1-p) = 5 * 2/10 * 8/10 = 0.8

The standard error can be calculated as:

SE = sqrt(Var(X)/n) = sqrt(0.8/5) = 0.4

(c) Yes, it would be appropriate to use the normal approximation for the number of winning plays since the number of trials is large enough (n=5) and the probability of success (p=2/10) is not too close to 0 or 1. We can assume that the number of winning plays follows a normal distribution with mean 1 and standard deviation 0.4.

Explanation: