Multiplying Probabilities Advanced Math Topics

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

Example 1 At a picnic, Julio reaches into an ice-filled cooler containing 8 regular soft drinks and 5 diet soft drinks. He removes a can, then decides he is not really thirsty, and puts in back. What is the probability that Julio and the next person to reach into the cooler both randomly select a regular soft drink? At a picnic, Julio reaches into an ice-filled cooler containing 8 regular soft drinks and 5 diet soft drinks. He removes a can, then decides he is not really thirsty, and puts in back. What is the probability that Julio and the next person to reach into the cooler both randomly select a regular soft drink? Ask yourself are these events independent? YES the outcome of the second person getting a drink is not affected by Julio’s selection Ask yourself are these events independent? YES the outcome of the second person getting a drink is not affected by Julio’s selection Make a plan, 13 cans and 8 are regular soft drinks so the probability of each person getting a regular soft drink is 8/13 Make a plan, 13 cans and 8 are regular soft drinks so the probability of each person getting a regular soft drink is 8/13 Solve, P(both regular soft drinks) = P(regular) ∙ P(regular) Solve, P(both regular soft drinks) = P(regular) ∙ P(regular) 8/13 ∙ 8/13 8/13 ∙ 8/13 = 38% = 38%

Example 2 Gerardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and replaces it. He then randomly selects another coin. What is the probability that both coins he selects are dimes? Gerardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and replaces it. He then randomly selects another coin. What is the probability that both coins he selects are dimes? Ask yourself are these events independent? YES, the outcome of the first selection doesn’t affect the second selection Ask yourself are these events independent? YES, the outcome of the first selection doesn’t affect the second selection Make a plan, 16 coins and 9 are dimes so the probability of picking a dime is 9/16 Make a plan, 16 coins and 9 are dimes so the probability of picking a dime is 9/16 Solve, P(Both dimes) = P(dime) ∙ P(dime) Solve, P(Both dimes) = P(dime) ∙ P(dime) 9/16 ∙ 9/16 9/16 ∙ 9/16 81/256 81/256

Example 3 In a board game, three dice are rolled to determine the number of moves for the players. What is the probability that the first die shows a 6, the second die shows a 6, and the third die does not? In a board game, three dice are rolled to determine the number of moves for the players. What is the probability that the first die shows a 6, the second die shows a 6, and the third die does not?

Example 4 When three dice are rolled, what is the probability that the first two show a 5 and the third shows an even number? When three dice are rolled, what is the probability that the first two show a 5 and the third shows an even number?

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

Back to Example 1 Part B What is the probability that both people select a regular soft drink if Julio does not put his back in the cooler? What is the probability that both people select a regular soft drink if Julio does not put his back in the cooler? Ask are these two events dependent, YES, the outcome of the second person choosing a drink is affected by Julio not putting his back Ask are these two events dependent, YES, the outcome of the second person choosing a drink is affected by Julio not putting his back Make a plan, Julio has a 8/13 probability while the second person now has 7/12 Make a plan, Julio has a 8/13 probability while the second person now has 7/12 Solve, 8/13 ∙ 7/12 = 56/156 = 14/39 Solve, 8/13 ∙ 7/12 = 56/156 = 14/39 = 36% = 36%

Example 5 In a state lottery game, each of three cages contains 10 balls. The balls are each labeled with one of the digits 0-9. What is the probability that the first two balls drawn will be even and that the third will be prime? In a state lottery game, each of three cages contains 10 balls. The balls are each labeled with one of the digits 0-9. What is the probability that the first two balls drawn will be even and that the third will be prime?

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

Example 6 part b Use the information above. What is the probability that the host draws two televisions? Use the information above. What is the probability that the host draws two televisions?

Example 7 The host of a game show is drawing ships from a bag to determine the prizes for which contestants will play. Of the 20 chips, of which 11 say computer, 8 say trip, and 1 says truck. If chips are drawn at random and without replacement, find the probability of drawing a computer, then a truck. The host of a game show is drawing ships from a bag to determine the prizes for which contestants will play. Of the 20 chips, of which 11 say computer, 8 say trip, and 1 says truck. If chips are drawn at random and without replacement, find the probability of drawing a computer, then a truck.

Example 8 Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a heart, another heart, and a spade in that order. Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a heart, another heart, and a spade in that order.

Example 9 Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a diamond, a club, and another diamond in that order. Three cards are drawn from a standard deck of cards without replacement. Find the probability of drawing a diamond, a club, and another diamond in that order.

Example 10 Find the probability of drawing three cards of the same suit. Find the probability of drawing three cards of the same suit.