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 likes McDonald’s or Burger King? 3. What is the probability that a randomly chosen person is female or likes McDonald’s? 65/100 = 13/20 McDonald’sBurger KingWendy’s Male Female /4

Math I UNIT QUESTION: How do you use probability to make plans and predict for the future? Standard: MM1D1-3 Today’s Question: When do I add or multiply when solving compound probabilities? Standard: MM1D2.a,b.

Probability Conditional Probability and Independent vs. Dependent events

Conditional Probability A conditional probability is the probability of an event occurring, given that another event has already occurred. The conditional probability of event B occurring, given that event A has occurred, is denoted by P(B/A) and is read as “probability of B, given A.” This is an “and” question, and solved by multiplication

Conditional Probability Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king, and we did not replace the king. Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king, and we did not replace the king. Because the first card is a king and is not replaced, the remaining deck has 51 cards, 4 of which are queens, so P(B/A) = 4/51  Because the first card is a king and is not replaced, the remaining deck has 51 cards, 4 of which are queens, so P(B/A) = 4/51 

Conditional Probability Gene Present Gene Not Present Total High IQ Normal IQ Total The above table shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ, given that the child has the gene. The above table shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ, given that the child has the gene.

Conditional Probability Gene Present Gene Not Present Total High IQ Normal IQ Total There are 72 children who have the gene so the sample space consists of these 72 children. Of these, 33 have a high IQ, so There are 72 children who have the gene so the sample space consists of these 72 children. Of these, 33 have a high IQ, so P(B/A) = 33/72  P(B/A) = 33/72  0.458

Conditional Probability Gene Present Gene Not Present Total High IQ Normal IQ Total Find the probability that a child does not have the gene. Find the probability that a child does not have the gene. P(child does not have the gene) = 30/102 P(child does not have the gene) = 30/102 Find the probability that a child does not have the gene, given that the child has a normal IQ Find the probability that a child does not have the gene, given that the child has a normal IQ P(B/A) = 11/50 P(B/A) = 11/50

What is the probability of the blood being type B given it is positive? What is the probability of the blood being type B given it is positive? 37/344 37/344 What is the probability of the blood being type RH Positive, given it is B or AB? What is the probability of the blood being type RH Positive, given it is B or AB? ( )/( ) = 49/61 ( )/( ) = 49/61 Blood Type OABABTotal RH Factor Positive Negative Total

Independent Events Two events A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Two events A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Then P(B/A) = P(B) Then P(B/A) = P(B) 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. 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: Probability of A and B occurring: P(A and B)=P(A)*P(B) P(A and B)=P(A)*P(B)

Probability NOTE: NOTE: You add something to get the probability of something OR something You add something to get the probability of something OR something You multiply something to get the probability of something AND something. You multiply something to get the probability of something AND something.

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. 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)=1/2 P (head)=1/2 P(3)=1/6 P(3)=1/6 P (head and 3)=P (head)*P(3) P (head and 3)=P (head)*P(3) =1/2 * 1/6 =1/2 * 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? 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)= 4/52 P (jack)= 4/52 P (8)= 4/52 P (8)= 4/52 P (jack and 8)= 4/52 * 4/52 P (jack and 8)= 4/52 * 4/52 = 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? 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) x P (yellow) P (green and yellow) = P (green) x 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? 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) = 9/10 P (student 1 likes pizza) = 9/10 P (student 2 likes pizza) = 9/10 P (student 2 likes pizza) = 9/10 P (student 3 likes pizza) = 9/10 P (student 3 likes pizza) = 9/10 P (student 1 and student 2 and student 3 like pizza) = 9/10 x 9/10 x 9/10 = 729/1000 P (student 1 and student 2 and student 3 like pizza) = 9/10 x 9/10 x 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. 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. 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: Probability of A and B occurring: P(A and B)=P(A)*P(B/A) P(A and B)=P(A)*P(B/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? 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) = 5/16 P (green) = 5/16 P (yellow given green) = 6/15 P (yellow given green) = 6/15 P (green and then yellow) = P (green) x P (yellow) P (green and then yellow) = P (green) x P (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? 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) = 6/10 P (male) = 6/10 P (male given 1 st male) = 5/9 P (male given 1 st male) = 5/9 P (male and then, male) = 1/3 P (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? 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 1 st bad) = 4/99 P (bad given 1 st bad) = 4/99 P (bad and then, bad) = 1/495 P (bad and then, bad) = 1/495

Class Work Pg 353, # 5 – 8 all and Pg 353, # 5 – 8 all and Handout: # 5-40 and 5-73 through 5-88 Handout: # 5-40 and 5-73 through 5-88