PROBABILITY

Questions, comments, concerns?Ok to move on?

Vocab

Trial- number of times an experiment is repeated

Outcomes- different results possible Frequency- number of times an event

occurs Relative Frequency- ??

Warm UpList all the possible outcomes for tossing a coin once…

Let an event be tossing a single coin two times. Represent the sample space :a. In a list form b. in a table form c. as a tree diagram.

Head, Tail or HT

HH, HT, TH, TT

Toss 1

Toss 2

H H

H T

T H

T T

Practice

An urn contains 5 purple, 3 white, and 4 red marbles. Two marbles are drawn. Draw a sample space for the event using a TREE DIAGRAM.

Practice

Let the event be a couple having children. Draw a tree diagram for the outcomes of a couple having four children.

Practice

Let the event be tossing a coin and then rolling a fair six-sided die. Draw a tree diagram and list the outcomes

Practice

The town of Alright has a population of 200 adults: 60% are women and 10% of them are more than 6ft tall. Of the men, 30% are more than 6 ft tall. Draw and label a tree diagram of the town.

NOW the juicy stuff… Probability!Probability:

Formula in set notation?

Theoretical vs Experimental Probability?

Practice

Find the probability that a 7 will show when rolling a fair six-sided die whose faces are numbered 1 through 6.

Practice

Find the probability that an even number shows when a six-sided die is rolled.

Review

A compliment is….

P(A’) = 1- P(A)

Ex: If there is a 35% chance it will rain tomorrow, what is the probability it will NOT rain.

Practice

Find the probability that when rolling two dice they will not show doubles.

Laws of Probability

Mutually Exclusive: P (A U B) = P(A) + P(B)Combined Events: P(A U B) = P(A) + P(B) – P(A∩ B)

Independent Event: P(A ∩ B) = P(A) ∙ P(B)Dependent Event: P (B|A) =

Note: OR = UnionAND = Intersection

Mutually Exclusive

If it is not possible for the events to occur at the same time.

Mutually Exclusive Probability P(A U B) = P(A) + P(B) where sets A and B have no elements in common.

Example

Let A be the event that a person is a female and B be the event that a person is 1.6 metres tall. a. Are the two mutually exclusive?b. Justify your answer by drawing a

venn diagram

Food for thought

Can two political parties be considered mutually exclusive?

Practice

Find the probability of having a 5 or an even number show when rolling a six-sided die.

Combined Events

These events are NOT mutually exclusive.

Combined Events ProbabilityP(A U B) = P(A) + P(B) – P(A ∩ B)

Consider the following: U= numbers 1-12. A = multiples of 3 numbers. B= Even numbers. Create a Venn diagram.

What is P(A U B)?

Practice

Find the probability, when drawing a single card from a standard 52-card deck, that a card is either a nine or a face card.

Practice

Find the probability, when drawing a single card from a 52-card deck, that a card is either red or an ace.

Practice

If P(A) = .37, P(B) = .55, P(A∩B) = .13, find: P(AUB) P(A’) P(B’) P(A∩B’)

Data was collected for which gender reads which types of book. A book as selected at random. Find the probability that:

a. The book was read by a female or it was an action book.

b. The book was a historical book or it was read by a male.

c. The book was not read by a male and it was an action book. Action Romance Historical

Male 10 17 8

Female 3 4 8

Independent Events

This occurs if the probability of the first event does NOT effect the probability of the second event…

Then to see what the probability of both happening is P(A∩B) = P (A) ∙ P(B)

Practice

If the probability it will snow on Monday is 25% and on Tuesday it is 40%, what is the probability that it will snow on both Monday and Tuesday?

Practice

Find the probability of the toss of a coin landing heads up and the roll of a die showing a four.

Practice

An urn contains 4 red marbles and 3 green marbles. Find the probability of drawing a red marble, replacing it, drawing a green marble, replacing it, and then drawing a red marble.

From a group of 20 athletes it is found that 13 played billiards, 12 played golf and 5 played both billiards and golf. a. Draw a venn diagram to represent this

information. b. Find the probability that an athlete chosen at

randoma. Plays golfb. Does not play billiardsc. Plays billiards and golfd. Plays billiards or golf

c. State a reason as to whether or not the events are independent.

d. State a reason as to whether or not the events are mutually exclusive.

Dependent Events

Two events are said to be dependent if the probability of the first event occurring influences the probability of the second event occuring.

Example

An urn contains 4 red marbles and 3 green marbles. Find the probability of drawing a red marble, NOT replacing it, and then drawing a green marble.

Practice

When drawing two cards from a standard 52-card deck, find the probability of both cards being black.

A bag contains two red sweets and three green sweets. Jacques takes one sweet from the bag, notes its colour, then eats it. He then takes another sweet from the bag. Copy and complete the tree diagram below to show all possibilities.

Conditional ProbabilityRecognizable by the words “given that”

P(B|A) =

Practice

Six identical marbles, numbered, 1,2,3,4,5,6 are placed in an urn. A single draw is made. It is known only that an even-numbered marble is selected. What is the probability that it is a four?

PracticeConsider the data collected:

a. Find the probability that the kind of vehicle driven to school was a truck, given that the student was a senior.

b. Given that the vehicle driven to school was a sports car, find the probability that the student was a junior.

Sports Car

Truck Sedan Totals

Junior 8 7 9 24

Senior 15 4 7 26

Totals 23 11 16 50

Practice

From a standard 52 card deck, let event D be the selection of a diamond and event F be the selection of a face card. One card is selected at random. Find the probability that:a. P(the card is a diamond or a face card)b. P(the card is neither a diamond nor a face card)c. P(a face card is selected given that a diamond has already been selected.)d. P(a diamond is selected given that a face card has already been selected. )