Vocabulary• 12.4 Concurrent Events

– Compound: two events that are considered together– Overlapping: events with one or more outcomes in common– Disjoint: events with no outcomes in common– Complement: Ā is the complement of A which is all

outcomes not in A

12.4 Probability of Compound Events

Compound Events• Compound: two events that are considered

together on one trial– Overlapping events have one or more outcomes in

common– Disjoint events have no outcomes in common (often

called mutually exclusive)

• Compound events will generally use the word ‘or’. (What is the probability of this or that happening when I …)

Overlapping Events• Overlapping events have one or more

outcomes in common

– Ex: a number thrown on a fair 6-sided die is < 4 or it is odd (1 and 3 satisfy both conditions)

– Ex: A card is drawn and it is a number (like 4) or a suit (like hearts)

Probability of Overlapping Events

• If A and B are two overlapping events, then the probability of A or B is:

P(A or B) = P(A) + P(B) – P(A and B)

= + -

(Each possibility must be accounted for once and only once)

A B

A & B

A B

SAT Test Practice

• A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a face card or a spade?

= + -

A. ³/₅₂ B. ¹⅟₂₆ C. ²⁵/₅₂ D. ⁷/₁₂

SpadeFace card

Face card Spade

Both

Deck of Cards Deck of Cards Deck of Cards Deck of Cards

Disjoint Events

• Disjoint events have no outcomes in common (often called mutually exclusive)

– Ex: A fair 12-sided die is rolled. What is the probability it will land with a 5 or a 9 up?

– Ex: a disk is drawn from a cup of 5 red, 6 blue, 4 green, and 5 yellow. What is the probability that the disk is yellow or green?

Probability of Disjoint Events

• If A and B are two disjoint events, then the probability of A or B is:P(A or B) = P(A) + P(B)

= +

(Each possibility must be accounted for once and only once)

A B A B

Find Probability of Disjoint Events

• A card is randomly selected from a standard deck of 52 cards, What is the probability that it is a 10 or a face card?

= +10

Face card 10

Face card

Deck of Cards Deck of CardsDeck of Cards

Use a formula to find P(A and B)

• Out of 200 students in a senior class, 113 students are either varsity athletes or on the honor roll. There are 74 seniors who are varsity athletes and 51 seniors who are on the honor roll. – What is the probability that a randomly selected senior is both a

varsity athlete and on the honor roll?

= + -Varsity Athlete

Honor Role

Honor Role

Varsity Athlete

Both

Seniors

Try these:A card is randomly drawn from a standard deck of 52 cards. Find the probability of the given event.

•Selecting an ace or an eight

•Selecting a 10 or a diamond

•In the senior class example, suppose 32 seniors are in band and 64 seniors are in the band or on the honor roll. What is the probability that a randomly selected senior is both in the band and on the honor roll?

Compliments• Ā is the compliment of A, which is

all outcomes not in A.

• The probability of Ā is: P(Ā) = 1 – P(A)

AĀ A is everything in

the pink circle.

Ā is everything in the blue region

Find Probabilities of compliments

When two six sided dice are rolled, there are 36 possible outcomes. Find the probability of:

a. The sum is not 6

P(sum not 6) = 1 – P(sum is 6)

b. The sum is less than or equal to 9

P(sum ≤ 9) = 1 –[P(sum is 10)+P(sum is 11)+P(sum

is 12)]

Use a Compliment in real life• A restaurant gives a free fortune cookie to every

guest. The restaurant claims there are 500 different messages hidden inside the fortune cookies. – What is the probability that a group of 5 people

receive at least 2 fortune cookies with the same message?

P(at least 2 fortunes match) = 1 – P(0

matches)

• HW 48: pg 727, 17-27 odd, 28-41 all