Page 95 - Probability Demystified

84 CHAPTER 5 Odds and Expectation

$2, $3, $4, $5, or $6; however, on average, you would win $3.50 on each roll.
So if you rolled the die 100 times, you would win on average
$3.50 100 ¼ $350. Now if you had to pay to play this game, you should
pay $3.50 for each roll. That would make the game fair. If you paid more to
play the game, say $4.00 each time you rolled the die, you would lose on
average $0.50 on each roll. If you paid $3.00 to play the game, you would win
an average $0.50 per roll.

EXAMPLE: When two coins are tossed, find the expected value for the
number of heads obtained.

SOLUTION:
Consider the sample space when two coins are tossed.

HH HT TH TT
j n = j
two heads one head zero heads
1
The probability of getting two heads is . The probability of getting one
4
1
1
1
1
head is þ ¼ . The probability of getting no heads is . The expected value
4 4 2 4
1
1
1
for the number of heads is EðXÞ¼ 2 þ 1 þ 0 ¼ 1:
4 2 4
Hence the average number of heads obtained on each toss of 2 coins is 1.
In order to find the expected value for a gambling game, multiply the amount
you win by the probability of winning that amount, and then multiply the
amount you lose by the probability of losing that amount, then add the results.
Winning amounts are positive and losses are negative.
EXAMPLE: One thousand raffle tickets are sold for a prize of an
entertainment center valued at $750. Find the expected value of the game
if a person buys one ticket.
SOLUTION:

The problem can be set up as follows:

Win Lose

Gain, (X ) $749 $1
1 999
Probability, P(X)
1000 1000