The Basics of Probability Theory

MATH 102Contemporary Math

S. Rook

Overview

• Section 14.1 in the textbook:– Sample spaces & events– Counting & probability– Odds

Sample Spaces & Events

Sample Spaces

• Random phenomena: an occurrence that varies between several different outcomes– e.g. Weather, rolling dice, flipping coins etc.

• An experiment is the observation of a random phenomena and noting the outcomes

• Sample space: the set of ALL possible outcomes for an experiment– e.g. List the sample space if we spin a spinner

with three regions: red, white, and blue

Probability & Events

• Probability is a measure of the likelihood of an outcome in the sample space occurring– The probability for an outcome is always between 0 and

1 inclusive• An outcome with a probability of 0 will NEVER occur• An outcome with a probability of 1 will ALWAYS occur

• An event is a subset of the possible outcomes in the sample space– The probability of an event, denoted P(E), is the sum of

the probabilities of the individual outcomes that make up the event

Sample Spaces & Events (Example)

Ex 1: A spinner has three regions: red, white, and blue. The spinner is spun twice and the results recorded.

a) List the sample spaceb) List those where red appears exactly once.c) List those outcomes where blue appears at least once.d) List those outcomes where yellow does not appear at all

Sample Spaces & Events (Example)

Ex 2: Suppose that we draw a card from a standard 52-card deck which is our sample space. List the event where we draw:

a) A 6 from the deck.

b) A red face card from the deck.

Counting & Probability

Counting & Probability

• Given that outcomes in the sample space are equally likely, the probability of event E is a where n(E) and n(S) represent the ardinal number of elements in the event set and sample space set respectively– For this class, each outcome in a sample space will

be equally likely– Possible that we may need the F.C.P., permutations,

or combinations to find n(E), n(S), or both

SnEn

EP

Counting & Probability (Example)

Ex 3: We are simultaneously rolling two four-sided dice having the numbers 1, 2, 3, and 4 on their faces. Outcomes in the sample space are listed as pairs such as (1, 2).

a) How many elements are in the sample space? b) What is the probability of the sum of the rolls being even?c) What is the probability that the sum of the rolls is greater than six?

Counting & Probability (Example)

Ex 4: Use the following table which relates living arrangements and GPA for 320 students:

a) If we select a random student, what is the probability that the student has a GPA of at least 2.5?b) If we select a random student, what is the probability that the student lives off-campus?

On Campus At Home Apartment Totals

Below 2.5 98 40 44 182

2.5 to 3.5 64 25 20 109

Over 3.5 17 4 8 29

Totals 179 69 72 320

Counting & Probability (Example)

Ex 5: In a given year, 2,048,861 males and 1,951,379 females were born in the U.S. If a child is selected randomly from this group, what is the probability that the child is a female?

Odds

Odds

• Odds is another commonly used concept that utilizes probability– Odds are often associated with betting

• e.g. If you win a bet having 6 : 1 odds, you will receive $6 for every $1 that you bet

• The odds against an event happening is the ratio of the probability of the event not happening and the probability of the event happening – Normally expressed in colon notation (a : b)– e.g. Consider the odds 6 : 1 again

• 6 out of 7 times event E would not happen

Odds (Continued)

• Consider a sample space with 10 outcomes– If 6 outcomes satisfy an event E, what is P(E)?– How many outcomes would NOT satisfy E? What is

this probability?– What are the odds against E?

Odds (Example)

Ex 6: If we draw one card from a standard 52-card deck:

a) What is the probability of drawing a spade?b) What are the odds against drawing a spade?

Odds (Example)

Ex 7: Solve:

a) If the odds against event E are 5 : 2, what is the probability of E?

b) What are the odds against the Yankees winning the World Series if the probability of them winning the world series is 0.30?

Summary

• After studying these slides, you should know how to do the following:– List and/or count the elements of the sample space of an

experiment– Calculate the probability P(E) of event E occurring– Calculate the odds of event E not occurring

• Additional Practice:– See problems in Section 14.1

• Next Lesson:– Complements & Unions of Events (Section 14.2)