Calculating expected values of roulette bets A roulette wheel has 38 numbers, 1 through 36, 0 and 00.  One-half of the numbers from 1-36 are red, and the other half are black; 0 and 00 are green.  A ball is rolled, and it falls into one of the 38 slots, giving a number and a color.  The payoffs (winnings) for a $1 bet are as follows:
Red or black    $1        0                   $35
Odd or even     $1        00                  $35
1-18            $1        any single number   $35
9-36            $1        0 or 00             $17

If a person bets \$1, find the expected value for each:
a) Red
b) Even
c) 00
d) Any single number
e) 0 or 00

The answers for each one is:  -$5.26. I need to know how to solve each one.

Progress: I know the expectation is to be the total of possible winnings times the odds of winning minus the amount expected to be lost times the odds of losing. But I'm not exactly sure how to calculate this equation because the amount actually won by obtaining a "red" is "zero"--the gambler would break even. However, by obtaining red, they would've lost the opportunities of the other possible winnings (I'm assuming), but I just don't end up with the answer that the book is looking for?
 A: You have only inflated your losses 100 times !
Computing for the first one, expected value = 2(18/38) - 1 = -\$0.0526 or -5.26 cents, or -5.26%
All your other parts can be computed similarly
A: Actually, I bit the bullet and began step-by-step help with Chegg (it actually coincides with my textbook and breaks down the problems one step at a time). The answer is:  E(x)=the sum of x * P(x)
for red:  the expected payout is \$1, with the odds of getting red 18/38; the payout for not getting red is -\$1, with the odds of not getting red 20/38.
So:  E(x)= (\$1)(18/38) + (-\$1)(20/38)
= 18/38 + (-20/38)
=-2/38
=-\$0.0526 or -5.26 cents
for even, it would be the same.
for "00": = (\$35)(1/38) + (-\$1)(37/38)
= 35/38 + (-37/38)
= -2/38
= \$0.0526 or -5.26 cents
for any single #: it would be the same
for "0 or 00": = (\$17)(2/38) + (-\$1)(36/38)
= 34/38 + (-36/38)
= -2/38
= -\$0.0526 or -5.26 cents
A: a.  $1\cdot (18/38) + (-1)\cdot(20/38)$
b.  $1\cdot (18/38) + (-1)\cdot(20/38)$
c.  $35\cdot (1/38) + (-1)\cdot(37/38)$
d.  $35\cdot (1/38) + (-1)\cdot(37/38)$
e.  $17\cdot (2/38) + (-1)\cdot(36/38)$
All values result to $-\$0.0526$ or an expected loss of $5.36$ cents
