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Counting and Gambling Section 4 Objectives: • Understand how to use the fundamental counting principle to analyze counting problems that occur in gambling. • Use the theory of permutations and combinations to compute the probability of drawing various card hands. Fundamental Counting Principle In how many different ways can the three wheels of the slot machine depicted below stop? Solution: Using the fundamental counting principle, we see that the total number of different ways the three wheels can stop is 20 × 20 × 20 = 8,000. Fundamental Counting Principle cont. (2) One payoff for a slot machine with the wheels shown below is for cherries on wheel 1 and no cherries on the other wheels. In how many different ways can this happen? Solution: By the fundamental counting principle, we see that the total number of ways we can get cherries only on the first wheel is 2 × 15 × 12 = 360. Fundamental Counting Principle cont. (3) The largest payoff for a slot machine with the wheels shown below is for three bars. In how many ways can three bars be obtained? Solution: By the fundamental counting principle, a bar on the first, second, and third wheel can occur in 2 × 3 × 1 = 6 ways. Counting and Poker In the game of poker, five cards are drawn from a standard 52-card deck. How many different poker hands are possible? Solution: Counting and Poker cont. (2) • Determine the number of different ways we can obtain four of a kind when drawing five cards from a standard 52-card deck. Solution: Step 1: Pick the rank of the card of which we will have four of a kind. There are 13 possibilities. Step 2: Select the fifth card. There are 48 cards remaining. Total: 13 × 48 = 624 ways. • How many ways can three cards of the same rank be drawn when drawing three cards from a standard 52-card deck? Solution: Step 1: Pick the rank of the card of which we will have three of a kind. There are 13 possibilities. Step 2: Choose three cards from the four cards in the deck with this rank. This can be done in C(4, 3) = 4 ways. Total: 13 × 4 = 52 ways Counting and Poker cont. (3) Determine the number of different ways we can obtain a full house when drawing five cards from a standard 52-card deck. Solution: Subproblem A: Determine the number of ways we can choose the three cards having the same rank. There are 52 ways from previous example. Subproblem B: Determine the number of ways we can choose the remaining two cards having the same rank. The rank can be chosen in only 12 ways. Selecting two of the four cards with this rank can be done in C(4, 2) = 6 ways. The two cards can be selected in 12 × 6 = 72 ways. Total: 52 × 72 = 3,744 ways. Counting and Poker cont. (4) If we select five cards from a standard 52-card deck, in how many ways can we draw a hand with three of a kind? Solution: Subproblem A: Determine the number of ways we can choose the three cards having the same rank. There are 52 ways from previous example. Subproblem B: Determine the number of ways we can choose the remaining two cards having the same rank. The fourth card can be chosen among the 12 remaining ranks in 48 ways, and then the fifth card can be chosen in 44 ways. To avoid counting rearrangements of these two cards, we divide by 2. The two cards can be selected in ways. Total: 52 × 1,056 = 54,912 ways