11.9 Solving Probability Problems by Using Combinations 739 In Section 11.5, we used the and probability formula to solve probability problems. Here we use the and formula to determine the probability of drawing 2 face cards, without replacement, from a standard 52-card deck. P P P (2 face cards) (first face card) (second face card) 12 52 11 51 3 13 11 51 33 663 11 221 = ⋅ = ⋅ = ⋅ = = The order in which the 2 cards are drawn does not matter, thus; this, problem can also be considered a combination problem. To determine the probability of drawing 2 face cards, we divide the number of different successful outcomes (drawing any 2 of the 12 face cards) by the number of total possible outcomes (drawing any 2 of the 52 cards in the deck). The number of ways in which two face cards can be selected from the 12 face cards in a deck is C , 12 2 or C 12! (12 2)!2! 12 11 10! 10! 2 1 66 12 2 6 = − = ⋅ ⋅ ⋅ ⋅ = The number of ways in which two cards can be selected from a standard 52-card deck is C , 52 2 or C 52! (52 2)!2! 52 51 50! 50! 2 1 1326 52 2 26 = − = ⋅ ⋅ ⋅ ⋅ = Thus, P C C (selecting 2 face cards) 66 1326 11 221 12 2 52 2 = = = Note that the same answer is obtained with either method. To give you more exposure to counting techniques, we will work the problems in this section using combinations. Fedor is playing poker and has been dealt a hand of 5 cards. What is the probability that Fedor has been dealt a royal flush? What is the probability that Fedor has been dealt a pair aces and a pair of 8’s? In this section, we will use combinations to answer questions such as these. Why This Is Important Probability problems that involve combinations are part of a branch of mathematics called combinatorics. Combinatorics has many applications in computer science, cyber security, engineering, and other branches of mathematics. Combinatorics is also used in the social sciences, genetics, and many business applications. Solving Probability Problems by Using Combinations SECTION 11.9 LEARNING GOAL Upon completion of this section, you will be able to: 7 Solve probability problems using combinations. Example 1 Math Club Committee The Math Club at Clackamas Community College has 5 freshmen and 7 sophomores. Joy, the club advisor, randomly selects 4 club members to attend the college’s interclub council. What is the probability that Joy selects 4 sophomores? HeidiBecker/Shutterstock
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