It is known that 10% of Year 12 students in a large country study advanced mathematics.A random sample of n students is selected from all Year 12's in this country, and the randomvariable X is the number of those in the sample who study advanced mathematics.(a) Describe the distribution of X. (1marks)(b) If they select 20 students, determine the probability thatNone of the students in the sample study advanced mathematics. (1 mark)If ten random samples of 20 students are selected, determine the probability that at least one ofthese samples has no students who study advanced mathematics. (2 marks)X 0 1 2 3𝑷(𝑿 = 𝒙) 𝟒𝒂𝟐𝟔𝑎2 − 𝑏2 𝑏2 0.5

(a) The distribution of X is a binomial distribution. This is because there are a fixed number of trials (n students), each trial is independent (the selection of one student does not affect the selection of another), and there are only two outcomes (the student either studies advanced mathematics or they do not). The probability of success (a student studying advanced mathematics) is constant (10%).

(b) If they select 20 students, the probability that none of the students in the sample study advanced mathematics can be calculated using the formula for the binomial probability:

P(X=0) = C(n, X) * (p^X) * ((1-p)^(n-X))

where:

• C(n, X) is the number of combinations of n items taken X at a time,
• p is the probability of success,
• n is the number of trials, and
• X is the number of successes.

Substituting the given values:

P(X=0) = C(20, 0) * (0.1^0) * ((1-0.1)^(20-0)) = 1 * 1 * (0.9^20) = 0.1216 (approximately)

So, the probability that none of the students in the sample study advanced mathematics is approximately 0.1216 or 12.16%.

If ten random samples of 20 students are selected, the probability that at least one of these samples has no students who study advanced mathematics can be calculated as the complement of the probability that all samples have at least one student who studies advanced mathematics.

Let's denote the event "at least one sample has no students who study advanced mathematics" as A and the event "all samples have at least one student who studies advanced mathematics" as B. We know that P(A) = 1 - P(B).

The probability that a sample of 20 students has at least one student who studies advanced mathematics is 1 - P(X=0) = 1 - 0.1216 = 0.8784.

Since the samples are independent, the probability that all ten samples have at least one student who studies advanced mathematics is (0.8784)^10 = 0.1486 (approximately).

Therefore, the probability that at least one of these samples has no students who study advanced mathematics is 1 - 0.1486 = 0.8514 or 85.14%.