Question

In a​ lottery, you bet on seven-digit number between 0000000 and 1111111. For a​ $1 bet, you win $500,000 if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are u=0.05 ​(that is,5 cents) and σ=200.00 Joe figures that if he plays enough times every​ day, eventually he will strike it​ rich, by the law of large numbers. Over the course of several​ years, he plays 1 million times. Let x denote his average winnings.

a. Find the mean and standard deviation of the sampling distribution of x

b. About how likely is it that Joe​'s average winnings exceed​ $1, the amount he paid to play each​ time? Use the central limit theorem to find an approximate answer.

Answer 1

(a)

the mean of the sampling distribution of = 0.05

the standard deviation of the sampling distribution of =

(b)

To find P( > 1.00):

Z = (1.00 - 0.05)/0.20

= 4.75

By Technology, Cumulative Area Under Standard Normal Curve = 0.99999898

So,

P( > 1.00):= 1 - 0.99999898 = 0.00000102

So,

Answer is:

0.00000102