Page 90 - Probability Demystified

CHAPTER 5 Odds and Expectation 79

EXAMPLE: Two coins are tossed; find the odds in favor of getting two heads

SOLUTION:
1
When two coins are tossed, there are four outcomes and PðHHÞ¼ : PðEÞ¼ 1
4
1
3
¼ ; hence,
4 4
1 1
PðE Þ 4 4
odds in favor of two heads ¼ ¼ ¼
1 PðE Þ 1 3
1
4 4
1
1 3 1 4 1
¼ ¼ ¼
4 4 4 1 3 3
The odds are 1 : 3.
EXAMPLE: Two dice are rolled; find the odds against getting a sum of 9.

SOLUTION:
There are 36 outcomes in the sample space and four ways to get a sum of 9.
1
8
1
Pðsum of 9Þ¼ 4 ¼ , PðEÞ¼ 1 ¼ : Hence,
36 9 9 9
8 8
PðE Þ 9 9
odds of not getting a sum of 9 ¼ ¼ ¼
1 PðE Þ 8 1
1
9 9
8 1 8 9 1 8
¼ ¼ ¼
9 9 9 1 1 1
The odds are 8 : 1.
If the odds in favor of an event occurring are A : B, then the odds against
the event occurring are B : A. For example, if the odds are 1 : 15 that an event
will occur, then the odds against the event occurring are 15 : 1.
Odds can also be expressed as
number of outcomes in favor of the event
odds in favor ¼
number of outcomes not in favor of the event

For example, if two coins are tossed, the odds in favor of getting two heads
were computed previously as 1 : 3. Notice that there is only one way to get
two heads (HH) and three ways of not getting two heads (HT, TH, TT);
hence the odds are 1 : 3.