Dr. Fowler AFM Unit 7-8

Probability

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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Classical (or theoretical) probability is used when each outcome in a sample space is equally likely to occur.

Classical Probability

Example: A die is rolled. Find the probability of Event A: rolling a 5.

There is one outcome in Event A: {5}

1 0.1676

P(A) = “Probability of Event A.”

P(Event) = the favorable number of outcomes the number of possible outcomes

5

Empirical (or statistical) probability is based on observations obtained from probability experiments. The empirical frequency of an event E is the relative frequency of event E.

Empirical Probability

Example:A travel agent determines that in every 50 reservations she makes, 12 will be for a cruise.What is the probability that the next reservation she makes will be for a cruise?

Frequency of Event ( ) Total frequency

EP E

12 0.2450

P(cruise) =

fn

Definition – a probability model will always have:1) positive values & 2) total values adding up to 1

E = All outcomes with 2 Boys & 1 Girl = { BBG, BGB, GBB } = 3 Total

Calculate the probability that in a 3 child family there are 2 boys and 1 girl. Assume equally likely outcomes.

S = 8 Total outcomes listed

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Example: You roll a die. Find the probability that you roll a number less than 3 or a 4.

The events are mutually

exclusive.P (roll a number less than 3 or roll a 4)

= P (number is less than 3) + P

(4)

2 1 3 0.56 6 6

The Addition Rule – Mutually Exclusive

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The Addition Rule – Not Mutually Exclusive

Example:A card is randomly selected from a deck of cards. Find the probability that the card is a Jack or the card is a heart. The events are not mutually exclusive because

the Jack of hearts can occur in both events.

P (select a Jack or select a heart)

= P (Jack) + P (heart) – P (Jack of

hearts)4 13 152 52 52

1652

0.308

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Complementary EventsThe complement of Event E is the set of all outcomes in the sample space that are not included in event E. (Denoted E′ and read “E prime.”)

P(E) + P (E′ ) = 1

Example: There are 5 red chips, 4 blue chips, and 6 white chips in a basket. Find the probability of randomly selecting a chip that is not blue.

P(E) = 1 – P (E′ ) P (E′ ) = 1 – P(E)

P (selecting a blue chip) 4 0.26715

P (not selecting a blue chip) 4 111 0.73315 15

On the local news the weather reporter stated that the probability of rain tomorrow is 30%. What is the probability that it will not rain?

no rain 1 rain 1 0.3 0.7 70%P P

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Example:A die is rolled and two coins are tossed. Find the probability of rolling a 5, and flipping two tails.

Multiplication Rule

P (rolling a 5) = 1.6

Whether or not the roll is a 5, P (Tail ) = so the events are independent.

1,2

P (5 and T and T ) = P (5)· P (T )· P (T ) 1 1 16 2 2

124

0.042

Excellent Job !!!Well Done

Stop NotesDo Worksheet

Introduction to Probability:https://www.youtube.com/watch?v=YWt_u5l_jHs

Consider the experiment of drawing a card from a standard 52 card deck of playing cards (ace through king, four suits). Let E represent the event “draw a heart” and let F represent the event “draw an ace”.

13 and 4n E n F

(a) The word "and" means intersection so and

is 1 (The ace of hearts)

n E F

n E F

(b)The word “OR” means Union, so n (E or F) = 16 (all 13 hearts + ace of clubs + ace of diamonds + ace of spades).

Consider the experiment of drawing a card from a standard 52 card deck of playing cards (ace through king, four suits). Let E represent the event “draw a heart” and let F represent the event “draw an ace”.

13 1(c)

52 4P E 4 1

52 13P F

1(d)

52

n E FP E F

n S

1n E F

16n E F

16 4(e)

52 13

n E FP E F

n S

Notice – sum of probabilities is 1

Notice – sum of probabilities is 1

What is the probability that in a group of 10 people at least 2 people have the same birthday? Assume that there are 365 days in a year.