The Intersection and Union of Events and Conditional Probability

The Intersection and Union of Sets

The Intersection and Union of Events

ExampleSet S is the set of integers from 1 through 100, inclusive. What is the probability that a numberrandomly selected from the set (a) is divisible by 3? (b) is divisible by 5?(c) is divisible by 3 and by 5? (d) is divisible by 3 or by 5?

Probability of the Intersection and the Union of Events

ExampleA card is drawn from a standard pack of 52 cards. What is the probability that the card (a) is a face card? (b) is a red card?(c) is both a face card and red? (d) is either a face card or red?

ExampleIn a class of 30 students, there are 17 girls and 13 boys. Five are A students, and three of these students are girls. If a student is chosen at random, what is the probability of choosing a girl or an A

ExampleA city survey found that 47% of teenagers have a part time job. The same survey found that 78% plan to attend college. If 10% neither have a part time job nor plan to attend college, what is the probability that the teenager chosen at random has a part time job and plans to attend college?

Mutually Exclusive Events

ExampleExperiment: Selecting a positive integer between 1 and 20.Event A: The number is even. Event B: The number is odd.Are the two events mutually exclusive?

ExampleExperiment: Selecting a positive integer between 1 and 20.Event A: The number is a multiple of 2. Event B: The number is a multiple of 5.Are the two events mutually exclusive?

Union and Intersection of Mutually Exclusive Events

ExampleExperiment: Drawing a card from a standard pack of 52 cards.Event A: Drawing a QueenEvent B: Drawing an Ace

Example

Experiment: Randomly draw one ball out of the bag contains 3 red balls, 5 white balls, and 4 green balls. Event A: Drawing a red ball Event B: Drawing a green ball

Independent and Dependent Events

Example

A bag contains 3 red balls and 4 green balls. You randomly draw two balls out of the bag. Determine whether the following events are independent or dependent.

(1) The events of drawing a red ball and drawing a green ball, with the first ball being replaced before the second drawing.

(2) The events of drawing a red ball and drawing a green ball, with the first ball not being replaced before the second drawing.

Conditional Probability

Conditional Probability

Example

A bag contains 3 red balls and 4 green balls. Find the following probabilities.

(1) The probability of drawing a red ball followed by a green ball, with the first ball not being replaced before

Example

A bag contains 3 red balls and 4 green balls. Find the following probabilities.

(2) The probability of drawing a red ball followed by a green ball, with the first ball being replaced before the

ExampleJayesh and Meena are giving their driving tests today. The probability of Jayesh passing his test is 1/4. The probability of both passing their tests is 1/6. What is the probability of Meena passing her test?A) 1/24B) 1/2C) 1/3D) 2/3E) 2/5

ExampleIn a shipment of 100 televisions, 6 are defective. If a person buys two televisions from that shipment, what is the probability that both are defective?

ExampleFrom a box containing 4 black and 6 white mice, three mice are randomly chosen. What is the probability that all three are black?(A) 8/125(B) 1/30(C) 2/5(D) 1/720(E) 3/10

Example

What is the probability that the product of two integers (not necessarily different) randomly selected from the numbers 1 through 20, inclusive, is odd?(A) 0(B) 1/4(C) 1/2(D) 2/3

Example

What is the probability that the sum of two integers (not necessarily different) randomly selected from the numbers 1 through 20, inclusive, is even?(A) 0(B) 1/4(C) 1/2(D) 2/3(E) 3/4