Question

For a multistate? lottery, the following probability distribution represents the cash prizes of the lottery with their corresponding probabilities. Complete parts? (a) through? (c) below.	x? (cash prize,? $)		?P(x)		Copy to Clipboard + Open in Excel +
	Grand prizeGrand prize			0.000000008390.00000000839
	?200,000			0.000000360.00000036
	?10,000			0.0000018710.000001871
	100			0.0001479450.000147945
	7			0.0054037740.005403774
	4			0.0064119730.006411973
	3			0.011546880.01154688
	0			0.976487188610.97648718861

?(a) If the grand prize is

?$12 comma 000 comma 00012,000,000?,

find and interpret the expected cash prize. If a ticket costs? $1, what is your expected profit from one? ticket?The expected cash prize is

?$0.30 0.30.

?(Round to the nearest cent as? needed.)

What is the correct interpretation of the expected cash? prize?

A.

On? average, you will win

?$0.300.30

per lottery ticket.

B.

You will win

?$0.300.30

on every lottery ticket.

C.

On? average, you will profit

?$0.300.30

per lottery ticket.The expected profit from one? $1 ticket is

?$negative 0.70 ?0.70.

?(b) To the nearest? million, how much should the grand prize be so that you can expect a? profit? Assume nobody else wins so that you do not have to share the grand prize.

?$nothing

?(c) Does the size of the grand prize affect your chance of? winning? Explain.

A.

?Yes, because your expected profit increases as the grand prize increases.

B.

?No, because the expected profit is always? $0 no matter what the grand prize is.

Answer 1

DATA IS NOT CLEAR. I AM EXPECTING THAT I AM TAKING THE CORRECT VALUES

(A)

Our expected value will be E[x] = x1p1+x2p2+....+xnpn
we have 8 values for x, so we will sum from 1 to 8
Substituting in our given values we have
(14,000,000)(0.00000000839)+(200,000)(0.00000036)+(10,000)(0.000001871)+(100)(0.000147945)+(7)(0.0054037740)+(4)(0.0064119730)+(3)(0.01154688)+(0)(0.97648718861)

=.11746+.072+.01871+.0147945+0.03783+.02565+..03464

= 0.321085
So we can calculate our expected value to be 0.3211

If the ticket cost is $1, we can find our expected profit by taking the expected value (how much we expect to win from entering) minus the ticket cost.
We then have $0.3211 - $1 = $-0.6789

you will win $0.3211 per ticket that you buy (not on EVERY ticket, just the average).

And this is not your profit. Since you paid $1 for the ticket, you are losing $0.6789 on average for every ticket you buy, so the profit is -$0.68

(B)  So let's first find the average profit of zero (breaking even).

So that would mean (Expected value) - 1 = 0 (you are subtracting 1 since you paid $1) The expected value is now, if we call the "grand prize" that we don't know the value of, G, then we want to solve

(expected value - 1) = (G*0.00000000839 + (200,000)(0.00000036)+(10,000)(0.000001871)+(100)(0.000147945)+(7)(0.0054037740)+(4)(0.0064119730)+(3)(0.01154688)+(0)(0.97648718861) - 1

and we want this to equal zero

0 = G*0.00000000839 + 0.2036 - 1

0 = G*0.00000000839- 0.7964

G=94,919,547.08=95,000,000

(C) The size of the prize does not affect your chances of winning, because the probabilities are set by the nature of this lottery