Probability

6.2 Assigning probabilities to Events

• Random experiment

– a random experiment is a process or course of action, whose outcome is uncertain.

• Examples

Experiment p

• Flip a coin • Record a statistics test marks • Measure the time to assemble a computer

Outcomes

Heads and Tails Numbers between 0 and 100 numbers from zero and above

6.2 Assigning probabilities to Events

• Performing the same random experiment repeatedly, may result in different outcomes, therefore, the best we can do is consider the probability of occurrence of a certain outcome. • To determine the probabilities we need to define p and list the possible outcomes first.

1

Sample Space

• Determining the outcomes.

– Build an exhaustive list of all possible outcomes. – Make sure the listed outcomes are mutually exclusive.

• A list of outcomes that meets the two conditions above is called a sample space.

Sample Space: S = {O1, O2,…,Ok}

O1 O2

Sample Space a sample space of a random experiment is a list of all possible outcomes of the experiment. The outcomes must be mutually exclusive and exhaustive.

Simple Events Event

The individual outcomes are called simple events. Simple events cannot be further decomposed into constituent outcomes.

Our objective simple events of one or more is to determine P(A), the probability that event A will occur.

An event is any collection

Assigning Probabilities

– Given a sample space S={O1,O2,…,Ok}, the following characteristics for the probability P(Oi) of the simple event Oi must hold:

1. 2. i 1

0 k P Oi P Oi

1 for each i 1

– Probability of an event: The probability P(A) of event A is the sum of the probabilities assigned to the simple events contained in A.

2

Approaches to Assigning Probabilities and Interpretation of Probability

• Approaches

– The classical approach (games of chance – coin toss) – The relative frequency approach (assign probabilities based on history of outcomes) – The subjective approach (we assign probabilities based on a degree of belief)

• Interpretation

– If a random experiment is repeated an infinite number of times, the relative frequency for any given outcome is the probability of this outcome.

6.2 Joint, Marginal, and Conditional Probability • We study methods to determine probabilities of events that result from combining other events in various ways. • There are several types of combinations and relationships between events: p

– – – – Intersection of events Union of events Dependent and independent events Complement event

Intersection

• The intersection of event A and B is the event that occurs when both A and B occur. • The intersection of events A and B is denoted by (A and B). B) • The joint probability of A and B is the probability of the intersection of A and B, which is denoted by P(A and B)

3

Intersection

• Example 6.1 (pg. 182)

– A potential investor examined the relationship between the performance of mutual funds and the school the fund manager earned his/her MBA. – The following table describes the joint probabilities.

Mutual fund Mutual fund doesn’t outperform the market outperform the market Top 20 MBA program Not top 20 MBA program

.11 .06

.29 .54

Intersection

• Example 6.1 (pg. 182)– continued

– The joint probability of

[mutual fund outperform…] and […from a top 20 …] = .11 – The joint probability of [mutual fund outperform…] and […not from a top 20 …] = .06

Mutual fund outperforms the market (B1) Top 20 MBA program (A1) P(A1 and B1)

Mutual fund doesn’t outperform the market (B2)

.11 .06

.29 .54

Not top 20 MBA program (A2)

Intersection

• Example 6.1 (pg. 182) – continued

– The joint probability of

P(A1 and B1)

[mutual fund outperform…] and […from a top 20 …] = .11 – The joint probability of [mutual fund outperform…] and […not from a top 20 …] = .06

P(A2 and B1)

Mutual fund outperforms the market (B1) Top 20 MBA program (A1)…