Page 92 - Probability Demystified

CHAPTER 5 Odds and Expectation 81

3. When three coins are tossed, there are three ways to get two tails and
a head. They are (TTH, THT, HTT), and there are eight outcomes in
the sample space. The odds in favor of getting two tails and a head are
3 3 1
3
5
8 ¼ 8 ¼ ¼ 3 8 ¼ 3
3 5 8 8 8 1 5 5
1
8 8
The odds are 3 : 5.
3 1
4. There are 3 even numbers out of 6 outcomes; hence, PðevenÞ¼ ¼ :
6 2
The odds in favor of an even number are

1 1
1
1
2 ¼ 2 ¼ ¼ 1 2 1 ¼ 1
1 1 2 2 2 1 1 1
1
2 2
The odds are 1:1.

5. There are six ways to get a sum of 7 and 36 outcomes in the sample
6
space. Hence, P(sum of 7) ¼ 36 ¼ 1 6 and P(not getting a sum of 7) ¼
1
5
1 ¼ : The odds against getting a sum of 7 are
6 6
5 5
5
1
6 ¼ 6 ¼ ¼ 5 6 1 ¼ 5
5 1 6 6 6 1 1 1
1
6 6
The odds are 5 : 1.
Previously it was shown that given the probability of an event, the odds in
favor of the event occurring or the odds against the event occurring can be
found. The opposite is also true. If you know the odds in favor of an event
occurring or the odds against an event occurring, you can find the probability
of the event occurring. If the odds in favor of an event occurring are A : B,
then the probability that the event will occur is PðE Þ¼ A :
AþB
If the odds against the event occurring are B : A, the probability that the
event will not occur is PðEÞ¼ B :
BþA
Note: Recall that PðEÞ is the probability that the event will not occur or
the probability of the complement of event E.