Question

use the fact that the mean of a geometric distribution is \\( \mu=\frac{1}{p} \\) and the variance is \\( \sigma^{2}=\frac{q}{p^{2}} \\).
a daily number lottery chooses two balls numbered 0 to 9. the probability of winning the lottery is \\( \frac{1}{100} \\). let \\( x \\) be the number of times you play the lottery before winning the first time.
(a) find the mean, variance, and standard deviation. (b) how many times would you expect to have to play the lottery before winning? it costs \\( \\$ 1 \\) to play and winners are paid \\( \\$ 1000 \\). would you expect to make or lose money playing this lottery? explain.
(a) the mean is (type an integer or a decimal.)
the variance is (type an integer or a decimal.)
the standard deviation is
(round to one decimal place as needed.)
(b) you can expect to play the game 100 times before winning.
would you expect to make or lose money playing this lottery? explain.
a. you would expect to make money on average you would win \\( \\$ 1000 \\) once in every times you play. so the net gain would be \\( \\$ \\)
b. you would expect to lose money on average you would win \\( \\$ 1000 \\) once in every times you play. so the net gain would be \\( \\$ \\)

Explanation:

Step1: Calculate the net gain

On average, you win $1000$ once in every $100$ times you play. The cost of playing $100$ times is $1\times100 = 100$ dollars.
The net gain is $1000-100=900$ dollars. But wait, let's check again.
The cost of playing $n$ times is $C = n\times1$ dollars. The expected number of plays $n=\mu = 100$. The winning amount is $1000$ dollars.
The net gain $G=1000 - 100\times1$.

Step2: Analyze profit or loss

Since $G = 1000-100=900>0$, you would expect to make money.

Answer:

A. You would expect to make money. On average you would win $1000$ once in every $100$ times you play. So the net gain would be $900$.