MA375: Solution to a homework problem from this week or last week's homework
Spring 2009, Prof. Walther
6.4 #6.
Expected value of the lottery ticket will be the amount you receive in earnings minus the cost of buying the ticket.
The amount you earn depends, of course, on whether or not you win the lottery.
By the definition of expected value, we know that your expected earnings equal
P(win the lottery) * (your earnings if you win the lottery) + P(don't win the lottery) * (your earnings if you don't win).
But remember, expected value of the ticket is expected earnings minus the cost of the ticket.
Here's what we know so far: If you win, you get $10,000,000. If you don't win, you get $0.
What is the probability of winning the lottery?
There is 1 correct set of 6 numbers that you could choose (that's the numerator) and C(50, 6) different possibilities (that's the denominator).
So P(win) = 1 / C(50, 6), and now we also know P(don't win) = 1 - P(win) = [C(50, 6) - 1] / C(50, 6).
Substitute all this back into your expected earnings, and then remember to subtract the cost of the lottery ticket...
You get $10,000,000 * [1 / C(50, 6)] + $0 * {[C(50, 6) - 1] / C(50, 6)} - $1 = -$.37
In other words, your lottery ticket's expected value is -37 cents.
