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How many possibilities are there for how they are matched up, assuming that in each game it does matter who has the white pieces (in a chess game, one player has the white pieces and the other player has the black pieces)? 32 Introduction to Probability 7. Two chess players, A and B, are going to play 7 games. Each game has three possible outcomes: a win for A (which is a loss for B), a draw (tie), and a loss for A (which is a win for B). 5 points, and a loss is worth 0 points. (a) How many possible outcomes for the individual games are there, such that overall player A ends up with 3 wins, 2 draws, and 2 losses?

Here are some examples of story proofs, which also serve as further examples of counting. 1 (Choosing the complement). For any nonnegative integers n and k with k  n, we have ✓ ◆ ✓ ◆ n n = . k n k This is easy to check algebraically (by writing the binomial coefficients in terms of factorials), but a story proof makes the result easier to understand intuitively. Story proof : Consider choosing a committee of size k in a group of n people. We know that there are nk possibilities. But another way to choose the committee is to specify which n k people are not on the committee; specifying who is on the committee determines who is not on the committee, and vice versa.

36 Introduction to Probability 32. (c) Suppose that there are 16 balls in total, and that the probability that the two balls are the same color is the same as the probability that they are di↵erent colors. What are r and g (list all possibilities)? s A random 5-card poker hand is dealt from a standard deck of cards. Find the probability of each of the following possibilities (in terms of binomial coefficients). (a) A flush (all 5 cards being of the same suit; do not count a royal flush, which is a flush with an ace, king, queen, jack, and 10).

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Bass - Probability Theory


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