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Instructor: Yeong-Chyuan Chung

In the previous section, we found the probability of an event of the simple events whose union is E E by using the probabilities . Sometimes, we can compute probabilities without having to consider simple events. Instead, we can use some properties of the probability function. Sometimes, we can also use Venn diagrams to help us compute probabilities.

Let S be the sample space of an experiment, and let Then the probability function P E and F has the following properties: be events of the experiment.

1.

P ( E ) ≥ 0.

2.

P ( S ) = 1.

3. If E and F are mutually exclusive, then P ( E ∪ F ) = P ( E ) + P ( F ).

4. (Addition rule) If E and F are any two events, then P ( E ∪ F ) = P ( E ) + P ( F ) − P ( E ∩ F ) .

5. (Complements) P ( E c ) = 1 − P ( E ).

Property 1 says that every event has a probability that is non-negative.

Property 2 comes from the fact that probabilities are numbers that do not exceed 1, and the fact that the sample space S contains all possible outcomes.

Properties 3 and 4 should look familiar from our discussion of sets in § 6.2. Recall that events are subsets of the sample space (which is the universal set). So from and F where § 6.2, we know that if E are disjoint sets (and thus mutually exclusive events), then n n ( E ∪ F ) = n ( E ) + n ( F ) denotes “the number of elements of”. Intuitively 1 , thinking of probabilities as relative frequencies, we obtain the respective probabilities by dividing through by n ( S ), thereby arriving at the equation stated in property 3. The equation stated in property 4 is obtained from the addition rule in § 6.2 in the same way.

Property 5 can be obtained from properties 2 and 3. This is because exclusive events, and their union is S .

E and E c are mutually 1 Actually this reasoning is only valid for uniform sample spaces and not for general sample spaces.

1

2 § 7.3 Rules of Probability

(Exercise 2 in the text)

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Find the probability of the event that the sum of the numbers is either 7 or 11.

(Exercise 6 in the text)

For the same experiment as above, find the probability of the event that the sum of the numbers is at least 4.

space S (Exercise 24 in the text) . Suppose P ( E ) = .

6 ,

Let P ( F ) = .

4 E and , and F be two events of an experiment with sample P ( E ∩ F ) = .

2 . Compute (a) P ( E ∪ F ) (b) P ( E c ) (c) P ( E c ∩ F )

§ 7.3 Rules of Probability 3

(part of Exercise 26 in the text)

Let S = { s 1 , s 2 , s 3 , s 4 , s 5 , s 6 } be the sample space associated with an experiment having the following probability distribution.

Outcome Probability If A = { s 1 , s 2 } and B = { s 1 , s 5 , s 6 } , find s 1 1 3 s 2 1 8 s 3 1 6 s 4 1 6 s 5 1 12 s 6 1 8 (a) P ( A ) , P ( B ) (b) P ( A ∩ B ) (c) P ( A c ∪ B c )

(Exercise 40 in the text)

Among 500 freshmen pursuing a business degree at a university, 320 are enrolled in an economics course, 225 are enrolled in a mathematics course, and 140 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is enrolled in (a) An economics and/or a mathematics course?

(b) Exactly one of these two courses?

(c) Neither an economics course nor a mathematics course?