slides_1_probability

# The number of possible hands where order does matter

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The number of possible hands where order does matter is 52 P 5 52 51 50 49 48. But we want to only count different hands, not different orderings. Given any 5 cards, they can be arranged in 5 4 3 2 1 5! ways. So the number of different hands is 52 51 50 49 48 5! 2,598,960. 95

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Generally, when we select k objects from n where we disregard order, we say we have a combination of length k . Sometimes this is written as n C k . The general formula is n C k n P k k ! n ! k ! n k ! , which is often written as n k n ! k ! n k ! and read as “ n choose k .” It is the number of combinations of n (distinct) objects chosen k at a time. 96
In the example of dealing a 5-card hand from a deck of 52 cards, the number of possible hands is 52 5 , which we already calculated. Remember, this is without regard to order. 97

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EXAMPLE : Consider the lottery example, where 6 different digits from 0 to 9 must be chosen but where one wins by matching the 6 digits regardless of the order chosen. So, this is an unorderd setup where selection is without replacement. We have 10 objects to choose from – the digits from 0 to 9 – and we are selecting 6 of them. Therefore, the number of possible combinations is 10 6 10! 6!4! 10 9 8 7 4 3 2 210 When order does not matter, the chance of winning is 1/210 compared with 1/ 10!/4! 1/151,200 if we have to get the order right. 98
We have shown how to fill in three of the four entries in the table, where we have n objects and select k if them: Without Replacement With Replacement Ordered n ! n k ! n k Unordered n k n k 1 k See Casella and Berger for the lower-right-hand entry. 99

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Often we need to apply the multiplication rule with the counting rules. EXAMPLE : Suppose a soccer team consists of 18 players: 4 forwards, 6 midfielders, 6 backers, and 2 goalies. The coach wants to field a team with 2 forwards, 4 midfielders, 4 backers, and (by rule) one goalie. How many different combinations of players are possible, assuming that, within a position, each player can play anywhere on the field (for example, a forward can play left forward or right forward)? 100
SOLUTION: The total number is obtained by multiplying the number of possible lineups within each position. So think of choosing each position as a “task”: 4 2 6 4 6 4 2 1 6 15 15 2 2,700. 101

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