A random sample of 4 pairs of twins will be chosen from the sample of 27. Let X be the number of twin pairs from the sample of 4 drawn whereby at least one twin in the pair has an IQ score greater than 88. Show that
P(X = 2) = 0.23 to 2 d.p.
Don't know where to start. Thanks in advance for any help.
To solve this problem, we can use the concept of probability and combinations. Let's break it down step by step:
Step 1: Find the total number of possible outcomes.
In this case, we are choosing 4 pairs of twins from a sample of 27, so the total number of possible outcomes is given by the combination formula, which is denoted as C(n, r).
The formula is: C(27, 4) = 27! / (4!(27 - 4)!).
Calculating this value:
C(27, 4) = 27! / (4!(27 - 4)!)
= (27 * 26 * 25 * 24) / (4 * 3 * 2 * 1).
= 17550.
Therefore, there are 17550 possible outcomes.
Step 2: Find the number of favorable outcomes.
We want X to be equal to 2, which means we have to choose exactly 2 twin pairs out of the 4 that have at least one twin with an IQ score greater than 88. To calculate the number of favorable outcomes, we need to consider the following:
- We have to choose 2 twin pairs out of the 4.
- Each twin pair has at least one twin with an IQ score greater than 88.
Let's break it down further:
- We have 4 twin pairs, and at least one twin in each pair must have an IQ score greater than 88. There are 13 twins with an IQ score greater than 88 in the sample of 27, so the number of ways to choose at least one twin from each pair is a combination of 13 twins taken 4 at a time: C(13,4).
- Once we have chosen one twin from each pair, we are left with 10 remaining twins with IQ scores greater than 88. We need to choose the other twin from each of the chosen pairs, which can be done in 2 ways (only one twin remaining).
Therefore, the total number of favorable outcomes is: C(13, 4) * 2.
Calculating this value:
C(13, 4) = 13! / (4!(13 - 4)!)
= (13 * 12 * 11 * 10) / (4 * 3 * 2 * 1).
= 715.
So, the number of favorable outcomes is 715 * 2 = 1430.
Step 3: Calculate the probability.
Now that we have the number of favorable outcomes and the total number of possible outcomes, we can calculate the probability using the formula:
P(X = 2) = (number of favorable outcomes) / (total number of possible outcomes)
= 1430 / 17550.
Calculating this value:
P(X = 2) = 1430 / 17550
≈ 0.0816.
Rounded to 2 decimal places, P(X = 2) is approximately 0.08.
Therefore, P(X = 2) = 0.08 to 2 decimal places.