There are ten Cortland, seven Gala and eight Macintosh apples in a basket. You pick
three apples from the basket.
a) What is the probability that three picked apples are all Gala or all Macintosh?
b) What is the probability that three picked apples are one Gala and two Macintosh?
c) What is the probability that three picked apples are one Cortland, one Gala and one
Macintosh?
d) Develop a probability distribution X for the number of Gala apples picked
plzz help
To solve this problem, we can use the concept of probability and combinations. Let's break down each part of the question.
a) What is the probability that three picked apples are all Gala or all Macintosh?
To find this probability, we need to calculate the probability of picking all Gala apples and the probability of picking all Macintosh apples.
Probability of picking all Gala apples:
The number of ways to choose 3 Gala apples from 7 Gala apples is given by the combination formula (7 choose 3):
C(7, 3) = 7! / (3! * (7-3)!) = 35
The total number of ways to choose any 3 apples from a total of 25 (10 Cortland + 7 Gala + 8 Macintosh) is given by the combination formula (25 choose 3):
C(25, 3) = 25! / (3! * (25-3)!) = 2,300
Therefore, the probability of picking all Gala apples is 35/2,300.
Probability of picking all Macintosh apples:
The number of ways to choose 3 Macintosh apples from 8 Macintosh apples is given by the combination formula (8 choose 3):
C(8, 3) = 8! / (3! * (8-3)!) = 56
Therefore, the probability of picking all Macintosh apples is 56/2,300.
To find the overall probability of picking either all Gala or all Macintosh apples, we sum the probabilities of these two events:
P(all Gala or all Macintosh) = P(all Gala) + P(all Macintosh)
= 35/2,300 + 56/2,300
= 91/2,300
= 0.0396 (rounded to four decimal places)
Therefore, the probability that three picked apples are all Gala or all Macintosh is approximately 0.0396.
b) What is the probability that three picked apples are one Gala and two Macintosh?
To find this probability, we need to calculate the probability of picking one Gala apple and two Macintosh apples.
Probability of picking one Gala and two Macintosh:
The number of ways to choose 1 Gala apple from 7 Gala apples is given by the combination formula (7 choose 1):
C(7, 1) = 7! / (1! * (7-1)!) = 7
The number of ways to choose 2 Macintosh apples from 8 Macintosh apples is given by the combination formula (8 choose 2):
C(8, 2) = 8! / (2! * (8-2)!) = 28
Therefore, the probability of picking one Gala and two Macintosh apples is (7 * 28)/2,300.
To calculate the probability, divide the number of favorable outcomes by the total number of possible outcomes:
P(one Gala and two Macintosh) = (7 * 28)/2,300
= 196/2,300
= 0.0852 (rounded to four decimal places)
Therefore, the probability that three picked apples are one Gala and two Macintosh is approximately 0.0852.
c) What is the probability that three picked apples are one Cortland, one Gala, and one Macintosh?
To find this probability, we need to calculate the probability of picking one Cortland apple, one Gala apple, and one Macintosh apple.
Probability of picking one Cortland, one Gala, and one Macintosh:
The number of ways to choose 1 Cortland apple from 10 Cortland apples is given by the combination formula (10 choose 1):
C(10, 1) = 10! / (1! * (10-1)!) = 10
The number of ways to choose 1 Gala apple from 7 Gala apples is given by the combination formula (7 choose 1):
C(7, 1) = 7! / (1! * (7-1)!) = 7
The number of ways to choose 1 Macintosh apple from 8 Macintosh apples is given by the combination formula (8 choose 1):
C(8, 1) = 8! / (1! * (8-1)!) = 8
Therefore, the probability of picking one Cortland, one Gala, and one Macintosh apple is (10 * 7 * 8)/2,300.
To calculate the probability, divide the number of favorable outcomes by the total number of possible outcomes:
P(one Cortland, one Gala, and one Macintosh) = (10 * 7 * 8)/2,300
= 560/2,300
= 0.2435 (rounded to four decimal places)
Therefore, the probability that three picked apples are one Cortland, one Gala, and one Macintosh is approximately 0.2435.
d) Develop a probability distribution X for the number of Gala apples picked.
To develop a probability distribution for the number of Gala apples picked, we need to calculate the probabilities for each possible outcome: picking 0, 1, 2, or 3 Gala apples.
Probability of picking 0 Gala apples:
The number of ways to choose 0 Gala apples from 7 Gala apples is given by the combination formula (7 choose 0):
C(7, 0) = 7! / (0! * (7-0)!) = 1
The number of ways to choose 3 apples from the remaining 18 (10 Cortland + 8 Macintosh) is given by the combination formula (18 choose 3):
C(18, 3) = 18! / (3! * (18-3)!) = 816
Therefore, the probability of picking 0 Gala apples is 1/816.
Similarly, you can calculate the probabilities for picking 1, 2, or 3 Gala apples.
Probability of picking 1 Gala apple:
Picking 1 Gala apple can occur in 7 ways (out of 7 Gala apples). The remaining 2 apples must be chosen from the other 18 (10 Cortland + 8 Macintosh) apples. So, the total number of outcomes is (7 * 18) = 126.
Therefore, the probability of picking 1 Gala apple is 126/816.
Probability of picking 2 Gala apples:
Picking 2 Gala apples can occur in (7 choose 2) ways. The remaining 1 apple must be chosen from the other 18 (10 Cortland + 8 Macintosh) apples. So, the total number of outcomes is C(7, 2) * 18 = 21 * 18 = 378.
Therefore, the probability of picking 2 Gala apples is 378/816.
Probability of picking 3 Gala apples:
This probability is similar to the probability of picking all Gala apples calculated in part a). So, the probability of picking 3 Gala apples is 35/2,300.
The developed probability distribution X for the number of Gala apples picked is as follows:
X = 0, 1, 2, 3
P(X) = 1/816, 126/816, 378/816, 35/2,300 (rounded to the appropriate fractions)
Please note that these values may be rounded for simplicity, but you can use the exact fractions for precise calculations.