Section 4.3 Rules of Probability

Properties of Probability Distributions

1. 0 P (si) 1 for i = 1, 2, · · · , n.

2. P (s1) + P (s2) + · · ·+ P (sn) = 1

3. P ({si} [ {si}) = P (si) + P (si) (i 6= j) for i = 1, 2, · · · , n; j = 1, 2, · · · , n.

1. Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with an experiment having the

following partial probability distribution.

Outcomes s1 s2 s3 s4 s5 s6

Probability 429

1029

229

329

229

Consider the events: A = {s1, s2, s5}, B = {s3, s5, s6}, C = {s1, s3, s4, s6}, and D = {s1, s2, s3}

Calculate the following probabilities. (Give answers as fractions.)

(a) P (s2)

(b) P (D)

(c) P (Bc)

(d) P (Ac \ B)

2. The sample space S = {s1, s2, s3} has the property that P (s1)+P (s2) = 0.39 and P (s2)+P (s3) =

0.83. Find the probability of each outcome.

Definition: Two events E and F are said to be mutually exclusive if the two events have no

outcomes in common, that is E \ F = ?.

Properties of Probabilities: Revisited

1. 0 P (E) 1 for any event E.

2. P (S) = 1.

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

4. If E and F are any two events of an experiment, then P (E[F ) = P (E)+P (F )�P (E\F ).

(This should remind us of the union/addition rule from section 6.2.)

5. If E is an event of an experiment and Ec denotes the complement of E, then P (Ec) =

1�P (E). The reverse is also true, namely, P (E) = 1�P (Ec). (I call this the Complement

Rule)

3. An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment

and find the probability of the event

A diamond or a king is drawn.

4. Let E and F be two events of an experiment with sample space S. Suppose P (E) = 0.59,

P (F ) = 0.38, and P (Ec \ F ) = 0.28. Calculate the probabilities below.

(a) P (Ec)

(b) P (E \ F )

(c) P (Ec \ F c)

(d) P (E [ F c)

2 Spring 2018, Maya Johnson

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5. Let E and F be two mutually exclusive events, and suppose P (E) = 0.6 and P (F ) = 0.1.

Compute the probabilities below.

(a) P (E \ F )

(b) P (E [ F )

(c) P (Ec)

(d) P (Ec \ F c)

(e) P (Ec [ F c)

6. Among 500 freshmen pursuing a business degree at a university, 317 are enrolled in an economics

course, 214 are enrolled in a mathematics course, and 138 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 each of the following? (Round answers to three decimal places.)

(a) an economics or a mathematics course

(b) exactly one of these two courses

(c) neither an economics course nor a mathematics course

3 Spring 2018, Maya Johnson

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7. A poll was conducted among 250 residents of a certain city regarding tougher gun-control laws.

The results of the poll are shown in the table. (Round answers to three decimal places.)

Own Own Own a

Only a Only a Handgun Own

Handgun Rifle and a Rifle Neither Total

Favor Tougher Laws 0 11 0 139 150

Oppose Tougher Laws 64 4 21 0 89

No Opinion 0 0 0 11 11

Total 64 15 21 150 250

(a) If one of the participants is selected at random, what is the probability that he or she opposes

tougher gun-control laws?

(b) If one of the participants is selected at random, what is the probability that he or she owns

a handgun but not a rifle?

(c) If one of the participants is selected at random, what is the probability that he or she favors

tougher gun-control laws and does not own a handgun?

4 Spring 2018, Maya Johnson

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