Question

In a raffle, you can pay $1 to select a number from 0 through 19. If you select the samenumber that is drawn, you winand collect $50 otherwise you get nothing.(a) How many different selections are possible?(b) What is the probability of winning?(c) If you win, what is your net profit?(d) Construct a probability distribution for this game.(e)Find the expected value E, of this raffle.

Answer 1

Hello there. To solve this question, we'll have to remember some properties about probabilities.

We know that, in a raffle, we pay $1 for choosing a number from 0 to 19. If we select the same number that is drawn, win and collect $50, otherwise get nothing.

a) How many different selections are possible?

Since we can choose only a number from 0 to 19, then we have 20 possible choices.

b) What is the probability of winning?

In this case, we have to take the same number that is drawn amongst the 20 possible, so the probability is the favorable event (taking the same card) divided by the total amount, 20.

This gives us

`P(win)=(1)/(20)`

c) If you win, what is your net profit?

In this case, since we have to pay $1 for choosing a number, we have a loss of $1 because of the chance of losing.

If we win, we get $50, then $50 - $1 = $49. This is the net profit in this case.

d) Construct a probability distribution for this game:

First, we'll construct it as the following table:

Since all the numbers have the same chance of being drawn, then this is the probability distribution for this game.

e) Find the expected value E, of this raffle

The expected value can be found by taking the sum of the values multiplied by their respective probability. Since they're all the same, as seen before, we have

`0\cdot(1)/(20)+1\cdot(1)/(20)+2\cdot(1)/(20)+\cdots+19\cdot(1)/(20)`

Factor the 1/20 term

`(0+1+2+\cdots+19)\cdot(1)/(20)`

Using the formula:

`1+2+\cdots+n=(n\cdot(n+1))/(2)`

We'll get

`(19\cdot20)/(2)\cdot(1)/(20)`

Cancelling the 20 term, the expected value will be:

`(19)/(2)=9.5`