Section 11.4:
Conditional Probability, Odds,
Expected Value
• Odds: Odds compares the number of favorable outcomes to the number of unfavorable
outcomes.
Suppose all outcomes in a sample space are equally likely where a of
them are favorable to the event E and the remaining b outcomes are
unfavorable to the event E.
* Odds in favor of E: are a to b, denoted a : b. In other words,
n(E) : n(E)
* Odds against: are b to a, denoted b : a. In other words,
n(E) : n(E)
• Odds ratio are normally simplified. For example, it is preferable to express odds as
12 : 1 rather than 48 : 4.
Example 1: A local baseball team has won 13 games and lost 2 games.
(a) What is the baseball team’s odds in favor of winning the next game?
(b) What is the baseball team’s odds against winning the next game?
Example 2: A card is drawn at random from a standard deck. Find
(a) Odds in favor of drawing a face card.
(b) Odds against drawing a diamond.
(c) Odds in favor of drawing the ace of spades.
(d) Odds against drawing a 2, 3 or 4.
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SECTION 11.4: CONDITIONAL PROBABILITY, ODDS, EXPECTED VALUE
Example 3: Suppose that the odds in favor of an event E are 3 : 7. Find P (E).
Example 4: Suppose that the odds in against an event E are 5 : 9. Find P (E).
Example 5: Suppose that P (E) =
4
. Find the odds against event E.
13
7
Example 6: Suppose that P (E) = . Find the odds in favor of event E.
9
Conditional Probability: Suppose A and B are events in a sample
space S such that P (B) 6= 0. The conditional probability that
event A occurs, given that event B occurs, denoted P (A|B), is
P (A|B) =
P (A ∩ B)
P (B)
Example 7: Suppose you draw one card from a fair deck of cards. What is the probability
that the card is a face card given that the card is a red card.
SECTION 11.4: CONDITIONAL PROBABILITY, ODDS, EXPECTED VALUE
3
Example 8: Given the spinner below, find the following probabilities.
8
1
7
2
6
3
5
4
(a) Probability that it lands on a five given that it lands on an odd number.
(b) Probability that it lands on a number less than 4 given that it lands on a shaded number.
(c) Probability that it lands on an odd number given that it lands on a shaded number.
(d) Probability that it lands on a shaded number given that it lands on an odd number.
(e) Probability that it lands on a 2 given that it lands on a factor of 10.
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SECTION 11.4: CONDITIONAL PROBABILITY, ODDS, EXPECTED VALUE
EXPECTED VALUE: Suppose that the outcomes of an experiment are real numbers called v1 , v2 , . . . , vn and suppose that the
outcomes have probability p1 , p2 , . . . , pn . The expected value of
the experiment is the sum
v1 · p1 + v2 · p2 + · · · + vn · pn
Stated in other words,
expected value = v1 · P (v1 ) + v2 · P (v2 ) + · · · + vn · P (vn ).
Example 9: An individual decides to play the following game. He tosses one fair die. He
loses $2 if he rolls a one; he loses $1 if he rolls a two or three; he breaks even if he rolls a four
or five; and he receives $5 if he rolls a six. Find the player’s expected winnings for this game.
Round your answer to two decimal places.
Example 10: According the a publisher’s records, 20% of the books published break even,
30% lose $1000, 25% lose $10, 000, and 25% earn $20, 000. When a book is published, what
is the expected income for the book?