A card is drawn from a standard deck of 52 cards. Find each probability. 1. P(a red card) ANSWER 1 2 2. P(a ten) ANSWER 1 13

3. What is the probability that two flipped coins show heads? ANSWER 1 4

EXAMPLE 1 Find probability of disjoint events A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a 10 or a face card? SOLUTION Let event A be selecting a 10 and event B be selecting a face card. A has 4 outcomes and B has 12 outcomes. Because A and B are disjoint, the probability is: P(A or B) = P(A) + P(B) = 4 52 + 12 16 0.308 13

EXAMPLE 2 Standardized Test Practice SOLUTION Let event A be selecting a face card and event B be selecting a spade. A has 12 outcomes and B has 13 outcomes. Of these, 3 outcomes are common to A and B. So, the probability of selecting a face card or a spade is:

EXAMPLE 2 Standardized Test Practice P(A or B) = P(A) + P(B) – P(A and B) = 12 52 + 13 3 – 52 22 = = 26 11 ANSWER The correct answer is B.

EXAMPLE 3 Use a formula to find P(A and B) Senior Class Out of 200 students in a senior class, 113 students are either varsity athletes or on the honor roll. There are 74 seniors who are varsity athletes and 51 seniors who are on the honor roll. What is the probability that a randomly selected senior is both a varsity athlete and on the honor roll? SOLUTION Let event A be selecting a senior who is a varsity athlete and event B be selecting a senior on the honor roll. From the given information you know that

Use a formula to find P(A and B) EXAMPLE 3 Use a formula to find P(A and B) 74 200 P(A) = , P(B) 51 , and P(A or B) 113 Find P( A and B ) P(A or B) = P(A) + P(B) – P(A and B) Write general formula. 113 200 74 = + 51 – P(A and B) Substitute known probabilities. P(A and B) 74 200 = + 51 – 113 Solve for P(A and B). P(A and B) 12 200 = 3 50 = 0.06 Simplify.

GUIDED PRACTICE for Examples 1, 2, and 3 A card is randomly selected from a standard deck of 52 cards. Find the probability of the given event. Selecting an ace or an eight ANSWER 2 13

GUIDED PRACTICE for Examples 1, 2, and 3 A card is randomly selected from a standard deck of 52 cards. Find the probability of the given event. Selecting a 10 or a diamond ANSWER 4 13

GUIDED PRACTICE for Examples 1, 2, and 3 What If? In Example 3, suppose 32 seniors are in the band and 64 seniors are in the band or on the honor roll. What is the probability that a randomly selected senior is both in the band and on the honor roll? 0.095 ANSWER

EXAMPLE 4 Find probabilities of complements Dice When two six-sided dice are rolled, there are 36 possible outcomes, as shown. Find the probability of the given event. The sum is not 6. The sum is less than or equal to 9.

EXAMPLE 4 Find probabilities of complements P(sum is not 6) = 1 – P(sum is 6) = 1 – 36 5 36 31 = 0.861 P(sum < 9) = 1 – P(sum > 9) = 1– 36 6 36 30 = 0.833 6 5 =

EXAMPLE 5 Use a complement in real life Fortune Cookies A restaurant gives a free fortune cookie to every guest. The restaurant claims there are 500 different messages hidden inside the fortune cookies. What is the probability that a group of 5 people receive at least 2 fortune cookies with the same message inside? SOLUTION The number of ways to give messages to the 5 people is 5005. The number of ways to give different messages to the 5 people is 500 499 498 497 496. So, the probability that at least 2 of the 5 people have the same message is:

EXAMPLE 5 Use a complement in real life P(at least 2 are the same) = 1 – P(none are the same) 500 499 498 497 496 = 1 – 5005 0.0199

GUIDED PRACTICE for Examples 4 and 5 Find P( A ). P(A) = 0.45 0.55 ANSWER

GUIDED PRACTICE for Examples 4 and 5 Find P( A ). P(A) = 4 1 ANSWER 4 3

GUIDED PRACTICE for Examples 4 and 5 Find P( A ). P(A) = 1 ANSWER

GUIDED PRACTICE for Examples 4 and 5 Find P( A ). P(A) = 0.03 0.97 ANSWER

GUIDED PRACTICE for Examples 4 and 5 WHAT IF? In Example 5, how does the answer change if there are only 100 different messages hidden inside the fortune cookies? ANSWER The probability increases to about 0.097.

Daily Homework Quiz 1. A card is randomly selected from a standard deck of 52 cards. What is the probability that it is a queen or an ace? ANSWER 2 13 2. Of 200 students at a school, 58 play football, 40 play basketball and 93 play both. What is the probability that a randomly selected student plays both? ANSWER 0.025

Daily Homework Quiz 3. When 2 six-sided dice are rolled, What is the probability that the sum is neither 2 nor 12? 17 18 ANSWER 4. A committee of 3 students is chosen from a group of 4 seniors and 6 juniors.What is the probability that the committee will have at least 1 senior? 56 ANSWER