The probability that a randomly chosen person in Botswana is 65 or older is approximately 0.2.
a) What is the probability that, in a randomly selected sample of 6 people, exactly 4 of them are 65 or older.
If X IS THE NUMBER of people of age 65 or older in a sample of 6, Construct the probability distribution of X
P(65+) = .2
P(<65) = .98
P(2<65 of 6) = .98^2 * .2^4 = ?
Cannot construct probability distribution here.
To find the probability that exactly 4 of the 6 people in the sample are 65 or older, we can use the binomial probability formula. The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials (sample size)
- k is the number of successes (in this case, the number of people 65 or older in the sample)
- p is the probability of success on each trial (in this case, the probability that a randomly chosen person is 65 or older)
In this case, the probability is 0.2, so p = 0.2. The sample size is 6 (n = 6). We want to find the probability of exactly 4 successes (k = 4).
Substituting these values into the formula, we get:
P(X = 4) = C(6, 4) * 0.2^4 * (1-0.2)^(6-4)
To find C(6, 4), we need to use the combination formula:
C(n, k) = n! / (k! * (n-k)!)
Substituting the values, we get:
C(6, 4) = 6! / (4! * (6-4)!) = 6! / (4! * 2!) = (6 * 5 * 4 * 3 * 2 * 1) / ((4 * 3 * 2 * 1) * (2 * 1)) = (6 * 5) / (2 * 1) = 15
Substituting this back into the original formula, we have:
P(X = 4) = 15 * 0.2^4 * (1-0.2)^(6-4)
Calculating this, we find:
P(X = 4) = 15 * 0.2^4 * 0.8^2 = 15 * 0.0016 * 0.64 = 0.01536
Therefore, the probability that exactly 4 of the 6 people in the sample are 65 or older is approximately 0.01536.
To construct the probability distribution for X (the number of people 65 or older in a sample of 6), you can use the same formula for each possible value of X from 0 to 6. Calculate the probability for each value of X and list them in a table. The sum of the probabilities should equal 1.