Answer :
The probability, we divide the total number of favorable outcomes by the total number of possible outcomes:
P = favorable outcomes / possible outcomes = (8008 + 17381 + 17381) / 12,271,512 ≈ 0.00219 or 0.219%.
(a) If he wants to serve 3 bottles of zinfandel and serving order is important, there are several ways to calculate the number of possible arrangements. Since there are 16 bottles of zinfandel, he can choose the first bottle in 16 ways. After choosing the first bottle, he will have 15 remaining options for the second bottle, and then 14 options for the third bottle. Therefore, the total number of ways to serve 3 bottles of zinfandel with serving order being important is calculated as:
16 * 15 * 14 = 3,360 ways.
(b) If 6 bottles of wine are to be randomly selected from the total supply of 48 bottles, we can use the concept of combinations to calculate the number of ways to do this. The formula for calculating combinations is given by nCr = n! / (r!(n-r)!), where n represents the total number of items and r represents the number of items to be chosen.
In this case, we have a total of 48 bottles and we want to select 6 bottles. Using the combination formula, we can calculate:
48C6 = 48! / (6!(48-6)!) = (48 * 47 * 46 * 45 * 44 * 43) / (6 * 5 * 4 * 3 * 2 * 1) = 12,271,512 ways.
(c) If he wants to randomly select 6 bottles and obtain two bottles of each variety (zinfandel, merlot, cabernet), we can break down the problem into selecting two bottles from each variety separately.
For zinfandel: There are a total of 16 zinfandel bottles, and we want to select two. Using the combination formula again:
16C2 = 16! / (2!(16-2)!) = (16 * 15) / (2 * 1) = 120 ways.
Similarly, for merlot and cabernet, there are 15 and 17 bottles respectively. Therefore, the number of ways to select two bottles from each variety is:
15C2 * 17C2 = (15! / (2!(15-2)!) * 17! / (2!(17-2)!)) = (15 * 14 / (2 * 1)) * (17 * 16 / (2 * 1)) = 3,150 ways.
(d) To calculate the probability of randomly selecting two bottles of each variety when choosing a total of 6 bottles, we need to determine the total number of favorable outcomes and divide it by the total number of possible outcomes.
The total number of favorable outcomes is calculated in part (c) as 3,150 ways.
The total number of possible outcomes is calculated in part (b) as 12,271,512 ways.
Therefore, the probability is:
P = favorable outcomes / possible outcomes = 3,150 / 12,271,512 ≈ 0.000256 or 0.0256%.
(e) If he wants to randomly select 6 bottles and have all of them be the same variety, we can calculate the probability by considering each variety separately.
For zinfandel: There are a total of 16 zinfandel bottles, and we want to select all 6. Using the combination formula:
16C6 = 16! / (6!(16-6)!) = (16 * 15 * 14 * 13 * 12 * 11) / (6 * 5 * 4 * 3 * 2 * 1) = 8008 ways.
Similarly, for merlot and cabernet, there are 15 and 17 bottles respectively. Therefore, the number of ways to select all 6 bottles from each variety is:
15C6 + 17C6 = (15! / (6!(15-6)!) + 17! / (6!(17-6)!)) = (15 * 14 * 13 * 12 * 11 * 10) / (6 * 5 * 4 * 3 * 2 * 1) + (17 * 16 * 15 * 14 * 13 * 12) / (6 * 5 * 4 * 3 * 2 * 1) = 5005 + 12376 = 17381 ways.
To calculate the probability, we divide the total number of favorable outcomes by the total number of possible outcomes:
P = favorable outcomes / possible outcomes = (8008 + 17381 + 17381) / 12,271,512 ≈ 0.00219 or 0.219%.
Learn more about the topic of Permutations and Combinations here:
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