Probability of Compound Events

compound event combines two or more events, using the word and or the word or. The word “or” in probability means Union  of two events The word “and” in probability means the intersection  of two events mutually exclusive have no common outcomes. P(A  B) = 0 Overlapping events have at least one common outcome. Vocabulary

The probability is found by summing the individual probabilities of the events: P(A  B) = P(A) + P(B) A Venn diagram is used to show mutually exclusive events. Mutually Exclusive Events

Find the probability that a girl’s favorite department store is Macy’s or Nordstrom. Find the probability that a girl’s favorite store is not JC Penney. Mutually Exclusive Events Macy’s0.25 Saks0.20 Nordstrom0.20 JC Penney0.10 Bloomingdale’s

When rolling two dice, what is probability that your sum will be 4 or 5? Possibilities sum of 4 _____________________________ Possibilities sum of 5 _____________________________ Total possible combinations of rolling 2 die ____________ P(sum4  sum 5) = P(sum5) + P(sum4) Mutually Exclusive Events 7/36 1&3, 2&2, 3&1 1&4, 2&3, 3&2, 4&1 36

What is the probability of picking a queen or an ace from a deck of cards Mutually Exclusive Events = 2/13 P(Ace) = 4/52 P(QN) = 4/52 P(AUQ) = 8/52

Probability that overlapping events A or B will occur expressed as: P(M  E) = P(M) + P(E) - P(M  E) Overlapping Events

Find the probability of picking a king or a club in a deck of cards. Overlapping Events Kings____ Clubs ____ Kings that are clubs ____ Total Cards ____ P(K  C) = P(K) + P(Clubs) – P(kings that are clubs) P(K  C) = 4/ /52 – 1/52 = 16/52= 4/13

Find the probability of picking a female or a person from Tennessee out of the 31 committee members. Overlapping Events FemMale TN84 AL63 GA73 Females ____ People from TN ____ Females from TM ____ Total People _____

Independent Events Two events A and B, are independent if A occurs & does not affect the probability of B occurring. Examples- Landing on heads from two different coins, rolling a 4 on a die, then rolling a 3 on a second roll of the die. Probability of A and B occurring: P(A and B) = P(A) ∙ P(B)

A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and a yellow marble? P (green) = 5/16 P (green) = 5/16 P (yellow) = 6/16 P (yellow) = 6/16 P (green and yellow) = P (green) P (yellow) P (green and yellow) = P (green) ∙ P (yellow) = 15 / 128 = 15 / 128

Dependent Events Two events A and B, are dependent if A occurs & affects the probability of B occurring. Examples- Picking a blue marble and then picking another blue marble if I don’t replace the first one. Probability of A and B occurring: P(A and B)=P(A) ∙ P(B given A)

A random sample of parts coming off a machine is done by an inspector. He found that 5 out of 100 parts are bad on average. If he were to do a new sample, what is the probability that he picks a bad part and then picks another bad part if he doesn’t replace the first? P (bad) = 5/100 P (bad) = 5/100 P (bad given bad) = 4/99 P (bad given bad) = 4/99 P (bad and then bad) = 1/495 P (bad and then bad) = 1/495

A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. A second marble is chosen. What is the probability of choosing a green and a yellow marble if the first marble is not replaced? P (green) = 5/16 P (green) = 5/16 P (yellow) = 6/15 P (yellow) = 6/15 P (green and yellow) = P (green) P (yellow) P (green and yellow) = P (green) ∙ P (yellow) = 30 / 240 = 1/8 = 30 / 240 = 1/8

A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. A second marble is chosen. What is the probability of choosing a green marble both times if the first marble is not replaced? P (green) = 5/16 P (green) = 5/16 P (green) = 4/15 P (green) = 4/15 P (green and green) = P (green) P (green) P (green and green) = P (green) ∙ P (green) = 20 / 240 = 1/12 = 20 / 240 = 1/12

P(A or B) = P(A) + P(B)P(A or B) = P(A) + P(B) - P(overlap) P(A and B) = P(A) ∙ P(B)P(A and B) = P(A) ∙ P(B given A) -Drawing a king or a queen -Selecting a male or a female -Selecting a blue or a red marble -Drawing a king or a diamond -rolling an even sum or a sum greater than 10 on two dice -Selecting a female from Georgia or a female from Atlanta WITH REPLACEMNT: -Drawing a king and a queen -Selecting a male and a female -Selecting a blue and a red marble WITHOUT REPLACEMENT: -Drawing a king and a queen -Selecting a male and a female -Selecting a blue and a red marble

Find Probabilities of Compound Events Example 1 Find the probability of A or B You randomly choose a card from a standard deck of 52 playing cards. Solution a.Choosing a 9 or a King are mutually exclusive events. a.Find the probability that you choose a 9 or a King. b.Find the probability that you choose an Ace or a spade.

Find Probabilities of Compound Events Example 1 Find the probability of A or B You randomly choose a card from a standard deck of 52 playing cards. Solution b.Because there is an Ace of spades, choosing an Ace or spade are ___________________. There are 4 Aces, 13 spades, and 1 Ace of spades. a.Find the probability that you choose a 9 or a King. overlapping events

Find Probabilities of Compound Events Example 2 Find the probability of A and B You roll two number cubes. What is the probability that you roll a 1 first and a 2 second? Solution The events are _____________. The number on one number cube does not affect the other. independent P(1) P(2)

Find Probabilities of Compound Events Example 3 Find the probability of A and B Markers A box contains 8 red markers and 3 blue markers. You choose one marker at random, do not replace it, then choose a second marker at random. What is the probability that both markers are blue? Solution Because you do not replace the first marker, the events are __________. Before you choose a marker, there are 11 markers, 3 of them are blue. After you choose a blue marker, there are 10 markers left and two of them are blue. So, the ______________________ that the second marker is blue given that the first marker is blue, is dependent 3 10 conditional probability

Find Probabilities of Compound Events Example 3 Find the probability of A and B Markers A box contains 8 red markers and 3 blue markers. You choose one marker at random, do not replace it, then choose a second marker at random. What is the probability that both markers are blue? Solution P(blue) P(blue given blue)

Find Probabilities of Compound Events 1.In a standard deck of cards, find the probability you randomly select a King of diamonds or a spade. Choosing a King of diamonds or a spade are mutually exclusive events.

Find Probabilities of Compound Events 2.In Example 3, suppose there are also 4 orange markers in the box. Calculate the probability of selecting a blue marker and then an orange marker, without replacement. P(blue) + P(orange given blue)