In this chapter we introduce the basics of probability.

Chapter 12From Randomness to Probability

Terminology

A random phenomenon is any activity in which what will happen is random.

Each observation of a random phenomenon is called a trial.

The value of the observation is called the outcome.

Terminologycontinued

The collection of all possible outcomes is called the sample space.

Any subset of the sample space is called an event.

Two events are called disjoint or mutually exclusive if they share no common outcomes.

Example 1

A coin is tossed repeatedly until either a “Heads” is achieved or the coin has been tossed 4 times without achieving a “heads”.

(a) List the sample space for this phenomenon.

(b) Let E = {coin tossed three times}. List the outcomes in E.

(c) Let F = {odd number of “Tails” are tossed}. List the outcomes in F.

Example 1continued

A coin is tossed repeatedly until either a “Heads” is achieved or the coin has been tossed 4 times without achieving a “heads”.

(d) Let G = {even number of tosses}. List the outcomes in G.

(e) Which, if any, of the above events are disjoint?

Theoretical Probability

If all outcomes in an event E are equally likely, the probability that event E occurs is:

Example 2

Suppose a single card is selected from a standard deck of playing cards. Find each of the following:

(a) P(6 is selected)

(b) P( is selected)

(c) P(card selected is a face card and black)

(d) P(card selected is a face card or black)

(e) P(card selected is a red queen or a black even number)

Probability Facts

For any event E,

If S is the sample space,

For any event E, the set of all outcomes not in E is called the compliment of E and is denoted EC

Probability Factscontinued

For disjoint events E and F,

E and F are independent if whether or not F occurs has nothing to do with whether of not E occurs.

For independent events E and F,

Example 3

In a bag are 7 blue ping pong balls and 3 white ping pong balls. We will select 3 balls from the bag (one at a time with replacement). Find the following probabilities.

(a) P(all are blue)

(b) P(exactly one ball is blue)

(c) P(at least 2 balls are blue)

(d) P(at least 1 ball is blue)

Example 4

In a bag are 7 blue ping pong balls and 3 white ping pong balls. We will select 3 balls from the bag (one at a time without replacement). Find the following probabilities.

(a) P(all are blue)

(b) P(exactly one ball is blue)

(c) P(at least 1 ball is blue)

Venn Diagrams

A Venn diagram is a graphical representation of a sample space and some event(s) from that sample space.

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4

E F 1 =  2 =  3 =  4 =

Example 5

A poll taken in a small community showed that 70% of the people enjoy watching football on TV, 64% enjoy watching baseball on TV, and 56% enjoy watching both. Draw a Venn diagram representing this sample space and two events.