Multiplying ProbabilitiesMultiplying Probabilities

Advanced Math TopicsAdvanced Math Topics

Probability of Two Independent Probability of Two Independent EventsEvents

If two events A and B are If two events A and B are independent then the probability of independent then the probability of both events occurring is P(A and B) = both events occurring is P(A and B) = P(A) P(A) ∙ P(B)∙ P(B) This can be applied to any number of This can be applied to any number of

independent events!independent events!

Example 1Example 1

At a picnic, Julio reaches into an ice-filled cooler At a picnic, Julio reaches into an ice-filled cooler containing 8 regular soft drinks and 5 diet soft drinks. containing 8 regular soft drinks and 5 diet soft drinks. He removes a can, then decides he is not really He removes a can, then decides he is not really thirsty, and puts in back. What is the probability that thirsty, and puts in back. What is the probability that Julio and the next person to reach into the cooler both Julio and the next person to reach into the cooler both randomly select a regular soft drink?randomly select a regular soft drink? Ask yourself are these events independent? YES the outcome Ask yourself are these events independent? YES the outcome

of the second person getting a drink is not affected by Julio’s of the second person getting a drink is not affected by Julio’s selectionselection

Make a plan, 13 cans and 8 are regular soft drinks so the Make a plan, 13 cans and 8 are regular soft drinks so the probability of each person getting a regular soft drink is 8/13probability of each person getting a regular soft drink is 8/13

Solve, P(both regular soft drinks) = P(regular) Solve, P(both regular soft drinks) = P(regular) ∙ P(regular)∙ P(regular) 8/13 ∙ 8/138/13 ∙ 8/13 = 38%= 38%

Example 2Example 2

Gerardo has 9 dimes and 7 pennies in his pocket. Gerardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and He randomly selects one coin, looks at it, and replaces it. He then randomly selects another replaces it. He then randomly selects another coin. What is the probability that both coins he coin. What is the probability that both coins he selects are dimes?selects are dimes? Ask yourself are these events independent? YES, the Ask yourself are these events independent? YES, the

outcome of the first selection doesn’t affect the second outcome of the first selection doesn’t affect the second selectionselection

Make a plan, 16 coins and 9 are dimes so the Make a plan, 16 coins and 9 are dimes so the probability of picking a dime is 9/16probability of picking a dime is 9/16

Solve, P(Both dimes) = P(dime) Solve, P(Both dimes) = P(dime) ∙ P(dime)∙ P(dime) 9/16 ∙ 9/169/16 ∙ 9/16 81/25681/256

Example 3Example 3

In a board game, three dice are In a board game, three dice are rolled to determine the number of rolled to determine the number of moves for the players. What is the moves for the players. What is the probability that the first die shows a probability that the first die shows a 6, the second die shows a 6, and the 6, the second die shows a 6, and the third die does not?third die does not?

Example 4Example 4

When three dice are rolled, what is When three dice are rolled, what is the probability that the first two show the probability that the first two show a 5 and the third shows an even a 5 and the third shows an even number?number?

Probability of Dependent Probability of Dependent EventsEvents

If two events A and B are dependent, If two events A and B are dependent, then the probability of both events then the probability of both events occurring is P(A and B) = P(A) occurring is P(A and B) = P(A) ∙ P(B ∙ P(B following A)following A)

Back to Example 1 Part BBack to Example 1 Part B

What is the probability that both people What is the probability that both people select a regular soft drink if Julio does not select a regular soft drink if Julio does not put his back in the cooler?put his back in the cooler? Ask are these two events dependent, YES, the Ask are these two events dependent, YES, the

outcome of the second person choosing a drink outcome of the second person choosing a drink is affected by Julio not putting his backis affected by Julio not putting his back

Make a plan, Julio has a 8/13 probability while Make a plan, Julio has a 8/13 probability while the second person now has 7/12the second person now has 7/12

Solve, 8/13 Solve, 8/13 ∙ 7/12 = 56/156 = 14/39∙ 7/12 = 56/156 = 14/39 = 36%= 36%

Example 5Example 5

In a state lottery game, each of three In a state lottery game, each of three cages contains 10 balls. The balls are cages contains 10 balls. The balls are each labeled with one of the digits 0-each labeled with one of the digits 0-9. What is the probability that the 9. What is the probability that the first two balls drawn will be even and first two balls drawn will be even and that the third will be prime?that the third will be prime?

Example 6Example 6

The host of a game show is drawing The host of a game show is drawing chips from a bag to determine the chips from a bag to determine the prizes for which contestants will play. Of prizes for which contestants will play. Of the 10 chips in the bag, 6 show the 10 chips in the bag, 6 show television, 3 show vacation, and 1 television, 3 show vacation, and 1 shows car. If the host draws the chips at shows car. If the host draws the chips at random and does not replace them, find random and does not replace them, find the probability that he draws a the probability that he draws a vacation, then a car. vacation, then a car.

Example 6 part bExample 6 part b

Use the information above. What is Use the information above. What is the probability that the host draws the probability that the host draws two televisions?two televisions?

Example 7Example 7

The host of a game show is drawing The host of a game show is drawing ships from a bag to determine the ships from a bag to determine the prizes for which contestants will play. prizes for which contestants will play. Of the 20 chips, of which 11 say Of the 20 chips, of which 11 say computer, 8 say trip, and 1 says truck. computer, 8 say trip, and 1 says truck. If chips are drawn at random and If chips are drawn at random and without replacement, find the without replacement, find the probability of drawing a computer, then probability of drawing a computer, then a truck.a truck.

Example 8Example 8

Three cards are drawn from a Three cards are drawn from a standard deck of cards without standard deck of cards without replacement. Find the probability of replacement. Find the probability of drawing a heart, another heart, and a drawing a heart, another heart, and a spade in that order.spade in that order.

Example 9Example 9

Three cards are drawn from a Three cards are drawn from a standard deck of cards without standard deck of cards without replacement. Find the probability of replacement. Find the probability of drawing a diamond, a club, and drawing a diamond, a club, and another diamond in that order.another diamond in that order.

Example 10Example 10

Find the probability of drawing three Find the probability of drawing three cards of the same suit.cards of the same suit.