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.

You'll get more from doing the assignment yourself than you will from copying somebody else's work.

Neil, you're a savage.

can u teach the class besides just reading the slides or nah lmk

To solve this problem, we need to understand the concept of probability and use it to calculate the likelihood of different outcomes.

We'll start by calculating the total number of possible ways to pick three apples from the basket, which is the combination of 25 apples taken 3 at a time:

n(total) = C(25, 3) = 25! / (3! * (25-3)!) = 25! / (3! * 22!) = (25 * 24 * 23) / (3 * 2 * 1) = 2300

a) Probability of picking three apples that are all Gala or all Macintosh:
To calculate this probability, we need to find the number of favorable outcomes (picking all Gala or all Macintosh apples) divided by the total number of possible outcomes.

Favorable outcomes: Either all Gala apples or all Macintosh apples.
n(favorable) = C(7, 3) + C(8, 3) = (7! / (3! * (7-3)!)) + (8! / (3! * (8-3)!)) = 35 + 56 = 91

Probability = n(favorable) / n(total) = 91 / 2300 = 0.0396 (rounded to four decimal places)

b) Probability of picking one Gala and two Macintosh apples:
Favorable outcomes: One Gala apple and two Macintosh apples.
n(favorable) = C(7, 1) * C(8, 2) = (7! / (1! * (7-1)!)) * (8! / (2! * (8-2)!)) = 7 * 28 = 196

Probability = n(favorable) / n(total) = 196 / 2300 = 0.0852 (rounded to four decimal places)

c) Probability of picking one Cortland, one Gala, and one Macintosh apple:
Favorable outcomes: One Cortland apple, one Gala apple, and one Macintosh apple.
n(favorable) = C(10, 1) * C(7, 1) * C(8, 1) = (10! / (1! * (10-1)!)) * (7! / (1! * (7-1)!)) * (8! / (1! * (8-1)!)) = 10 * 7 * 8 = 560

Probability = n(favorable) / n(total) = 560 / 2300 = 0.2435 (rounded to four decimal places)

d) Probability distribution X for the number of Gala apples picked:
To find the probability distribution, we calculate the probability of picking a specific number of Gala apples (ranging from 0 to 3).

Number of Gala apples picked (X) | Probability
----------------------------------------------
0 | C(18, 3) / n(total) = 816 / 2300 ≈ 0.3548
1 | C(7, 1) * C(18, 2) / n(total) = 378 / 2300 ≈ 0.1643
2 | C(7, 2) * C(18, 1) / n(total) = 126 / 2300 ≈ 0.0548
3 | C(7, 3) / n(total) = 35 / 2300 ≈ 0.0152

Note: C(n, r) represents the combination of n items taken r at a time.

This probability distribution gives us the likelihood of picking a specific number of Gala apples from the basket of apples.