Lottery expected value problem
 

Let X be your payoff from one play of the Lottery game. It costs $1.00 to play the game so if you lose then x = -1. To play Powerball you select 5 distinct numbers from 53 white numbers and 1 additional number from 42 red numbers. The state then randomly selects 5 distinct white numbers and 1 red number. For this problem we will assume that the Jackpot will pay $10,000,000.
The possible payouts are:
( a ) Match all 5 white and the red - pays $10,000,000
( b ) Match all 5 white only - pays $100,000
( c ) Match 4 white and the red - pays $5,000
( d ) Match 4 white only - pays $100
( e ) Match 3 white and the red - pays $100
( f ) Match 3 white only - pays $7
( g ) Match 2 white and the red - pays $7
( h ) Match 1 white and the red - pays $4

Note: When the Lottery claims you have won $4 actually you have won $3 since they do not return the dollar you paid to play. Remember this in your calculations.

Find the expected value of X.

I don't think I did this problem right, can anyone show me the correct way to go about this problem? Thanks in advance for any help.

SGS

Re: Lottery expected value problem
 

Does this look anywhere near correct for just the probability of each? For some reason this feels entirely wrong to me. My brain is hurting from a long night of irish whiskey.

A.) 1/C(53,5) * 1/C(42,1)
B.) 1/C(53,5)
C.) 49/C(53,5) * 1/C(42,1)
D.) 49/C(53,5)
E.) 2450/C(53,5) * 1/C(42,1)
F.) 2450/C(53,5)
G.) 124950/C(53,5) * 1/C(42,1)
H.) 6497400/C(53,5) * 1/C(42,1)

Re: Lottery expected value problem
 

.The possible payouts are:
( a ) Match all 5 white and the red - pays $10,000,000
1/C(53,5) * 42

~8 cents

( b ) Match all 5 white only - pays $100,000
41/C(53,5)* 42

~3cents

( c ) Match 4 white and the red - pays $5,000
5*48/C(53,5)*42

( d ) Match 4 white only - pays $100
5*48*41/C(53,5)*42

( e ) Match 3 white and the red - pays $100
C(5,3)*C(48,2)/C(53,5)*42

( f ) Match 3 white only - pays $7
C(5,3)*C(48,2)*41/C(53,5)*42

( g ) Match 2 white and the red - pays $7
C(5,2)*C(48,3)/C(53,5)*42

( h ) Match 1 white and the red - pays $4
5*C(48,4)/C(53,5)*42

~3cents


Boy these are terrible odds!!

Re: Lottery expected value problem
 

The other posters have all shown the correct setup.

One detail noone has yet mentioned is that if the top prize is hit by more than one ticket the prize is split (while most if not all of the other prizes are fixed amounts and they pay everyone who hits the same amount.)

Assuming all combinations of lottery numbers are equally likely to be played, this means the value of the top prize is not 10,000,000, but 10000000/(1+(# tickets sold this week)/120526770)), which is going to put a significant dent in your EV since several million tickets get sold every week.