Question

A friend of mine is giving a dinner party. His current wine supply includes 12 bottles of zinfandel, 8 of merlot, and 10 of cabernet (he only drinks red wine), all from different wineries.

(a) If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this?
ways

(b) If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this?
ways

(c) If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?
ways

(d) If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? (Round your answer to three decimal places.)


(e) If 6 bottles are randomly selected, what is the probability that all of them are the same variety? (Round your answer to three decimal places.)

Answer 1

Given data :- Number of Zinfandel bottles is = 12

Number of Merlot bottles is = 8

Number of Cabernet bottles is = 10

Total number of bottles is = 30

(a) If he wants to serve 3 bottles of zinfandel and serving order is important, then the number of ways to do this are

12C3 = 220 ways.

(b) If 6 bottles of wine are to be randomly selected from the 30 for serving, then the number of ways to do this are

30C6 = 593775 ways.

(c) If 6 bottles are randomly selected, then the number of ways that 2 bottles are obtained of each variety are

12C2 * 8C2 * 10C2 = 66 * 28 * 45 = 83160 ways.

(d) If 6 bottles are randomly selected, the probability that this results in two bottles of each variety being chosen is

( 12C2 * 8C2 * 10C2 ) / 30C6 = 83160/593775 = 0.140053 = 0.141. The probability is 14.10 % that this results in two bottles of each variety being chosen.

(e) If 6 bottles are randomly selected, then the probability that all of them are the same variety is

( 12C6 + 8C6 + 10C6 ) / 30C6 = ( 924+28+210) / 593775 = 0.001956

Therefore the probability is 0.195 % that all of them are of same variety.