7/20

The following table shows the number of people that like a particular fast food restaurant.

1) What is the probability that a person likes Wendy’s?

2) What is the probability that a person is male given they like Burger King?

3. What is the probability that a randomly chosen person is female or likes McDonald’s?

3/5

McDonald’s Burger King

Wendy’s

Male 20 15 10

Female 20 10 25

3/4

Probability

Independent vs. Dependent

events

Independent EventsIndependent Events• Two events A and B, are independent

if the fact that 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)

Experiment 1• A coin is tossed and a 6-sided die

is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die.

P (head) = 1/2P (head) = 1/2P(3) = 1/6P(3) = 1/6P (head and 3) = P (head) P (head and 3) = P (head) P(3) P(3)

= 1/2 = 1/2 1/6 1/6

= 1/12= 1/12

Experiment 2

• A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and an eight?P (jack) = 4/52P (jack) = 4/52

P (8) = 4/52P (8) = 4/52P (jack and 8) = 4/52 P (jack and 8) = 4/52 4/524/52

= 1/169= 1/169

Experiment 3

• 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) = 5/16P (green) = 5/16P (yellow) = 6/16P (yellow) = 6/16P (green and yellow) = P (green) P (green and yellow) = P (green) P (yellow) P (yellow)

= 15 / 128= 15 / 128

Experiment 4

• A school survey found that 9 out of 10 students like pizza. If three students are chosen at random with replacement, what is the probability that all three students like pizza?P (student 1 likes pizza) = 9/10P (student 1 likes pizza) = 9/10

P (student 2 likes pizza) = 9/10P (student 2 likes pizza) = 9/10P (student 3 likes pizza) = 9/10P (student 3 likes pizza) = 9/10P (student 1 and student 2 and student 3 P (student 1 and student 2 and student 3 like pizza) = 9/10 like pizza) = 9/10 9/10 9/10 9/10 = 729/1000 9/10 = 729/1000

Dependent Events• Two events A and B, are dependent if

the fact that 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)

Experiment 1

• 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 without replacing the first one. What is the probability of choosing a green and a yellow marble?

P (green) = 5/16P (green) = 5/16P (yellow given green) = 6/15P (yellow given green) = 6/15P (green and then yellow) = P (green) P (green and then yellow) = P (green) P P (yellow)(yellow)

= 1/8= 1/8

Experiment 2

• An aquarium contains 6 male goldfish and 4 female goldfish. You randomly select a fish from the tank, do not replace it, and then randomly select a second fish. What is the probability that both fish are male?

P (male) = 6/10P (male) = 6/10P (male given 1P (male given 1stst male) = 5/9 male) = 5/9P (male and then, male) = 1/3P (male and then, male) = 1/3

Experiment 3

• 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) = 5/100P (bad) = 5/100P (bad given 1P (bad given 1stst bad) = 4/99 bad) = 4/99P (bad and then, bad) = 1/495P (bad and then, bad) = 1/495

Independent vs. Dependent

Determining if 2 events are

independent

Independent EventsIndependent Events• Two events are independent if the

following are true:P(A|B) = P(A)  P(B|A) = P(B)  P(A AND B) = P(A) ⋅ P(B)

• To show 2 events are independent, you must prove one of the above conditions.

Experiment 1• Let event G  = taking a math class.

Let event H  = taking a science class. Then, G AND H  = taking a math class and a science class.

• Suppose P(G) = 0.6, P(H) = 0.5, and P(G AND H) = 0.3.• Are G and H independent?

( ) ( ) ( )?P G H P G P H

0.3 0.6 0.5

Experiment 2• In a particular college class, 60% of the

students are female. 50% of all students in the class have long hair. 45% of the students are female and have long hair. Of the female students, 75% have long hair.

• Let F be the event that the student is female. Let L be the event that the student has long hair.

• One student is picked randomly. Are the events of being female and having long hair independent?

( ) ( ) ( )?P F L P F P L

0.45 0.6 0.5

0.45 0.3