At some point during the season the Jays (baseball team) are going to go on a 7-game losing streak (It will happen). In order to break this streak, John Farrel decides to shake up the line a bit. Each name of the 13 batters is placed in a jar and is randomly drawn by him, recording the names of a 9-man batting order in the order in which they are drawn.
a)What is the probability that the new batter line-up is identical to the older one?
b)How many different line ups are there that have Edwin Encarnacion (a batter) batting in the 4th?
c) What is the probability that J.P Arenchiba (Another batter) will NOT be in the lineup?
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There are 13 players to choose from in order to create a 9-man roster. Since we are creating a batting order, order matters so we use permutations.
(1) The probability that the new order is the same as the old order:
There are `_13P_9=259459200` different batting orders to choose from. (Another way to see this: there are 13 players to choose from for slot 1, 12 players to choose from for slot 2, etc... or 13*12*11*10*9*8*7*6*5=259459200 different choices)
The probability of selecting the same batting order is `1/259459200~~3.85"x"10^(-9)`
(2) With a particular batter in the cleanup spot:
There are 12 players left to choose from and 8 slots to fill; again order matters:
So there are 19958400 different lineups with Edwin Encarnacion batting 4th.
(Again you can think that there are 12 men for the 1st slot, 11 for the 2nd, 10 for the 3rd, 9 for the 5th, 8 for the 6th, 7 for the 7th, 6 for the 8th, and 5 for the 9th or 12*11*10*9*8*7*6*5=19958400.)
(3) Probability that a particular batter is not in the lineup:
We found the number of potential lineups with all 13 available as `_13P_9=259459200` . The number of lineups choosing from 12 players is `_12P_9=79833600` .
The percentage of lineups that do not include J.P Arenchiba is `79833600/259459200~~.3077` or approximately 30.8%.
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