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Math 166, Fall 2015, Robert
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1.5- Rules for Probability
For any events E and F in a sample space S, we have
The Elementary Rules for Probability:
0 ≤ P (E) ≤ 1
P (S) = 1
P (∅) = 0
The Complement Rule for Probability:
P (E C ) = 1 − P (E)
P (E) = 1 − P (E C )
The Union Rule for Probability:
P (E ∪ F ) = P (E) + P (F ) − P (E ∩ F )
If E and F are mutually exclusive events, then P (E ∪ F ) =
Example 1 A die is loaded so that the probability that the probability that 1
comes up is 0.2, the probability that 2 comes up is 0.2, and the probability that 3
comes up is 0.3. The last three numbers are all equally likely to come up. What
is the probability that a 6 is rolled?
Example 2 A single card is drawn from a standard deck of cards. What is the
probability that the card is an ace or a spade?
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Math 166, Fall 2015, Robert
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Example 3 Two dice, one purple and one green, are rolled. Let E be the event
that the green die is even and F be the event that the sum of the dice is greater
than 8. What is P (E ∪ F )?
Example 4 The following data can be found on the CDC’s website for the
number of people who died in the US. in 2013 (we are disregarding the 132
deaths where age was not recorded).
Age
Number of Deaths
Under 1
23, 440
At least 1 but under 20
18, 888
At least 20 but under 30
40, 062
At least 30 but under 50
160, 813
At least 50 but under 70
663, 527
70 or older
1, 690, 131
Total Number of Deaths
2, 596, 861
What is the (empirical) probability that someone who died in 2013 was under
20? Under 70?
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Math 166, Fall 2015, Robert
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The odds in favor of an event E is the ratio of P (E) to P (E C ).
The ratio is often reduced to lowest terms, a/b, and we say that the odds are
a to b or a : b.
If the odds of E happening are a : b, then
P (E) =
Example 5 You’ve been hired to handle the bets taken at a local race track.
Damascus is estimated to have a probability of 0.6 to win the race.
1. What is the probability that Damascus will lose?
2. What are the odds of Damascus winning?
3. If someone bets $12 on Damascus and the horse wins, how much profit
should be given to be fair?
Example 6 After a clinical trial, the odds of suffereing several side effects are
estimated. Determine the probability of suffering each side effect.
1. headache 11 : 7
2. nausea 5 : 13
3. dizziness 2 : 11
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