Solving Probability Problems with Discrete Distributions

[chrisyuen]

Homework Statement

At a school sports day, the timekeeping group for running events consists of 1 chief judge, 1 referee and 10 timekeepers. The chief judge and the referee are chosen from 5 teachers while the 10 timekeepers are selected from 16 students.

(a) How many different timekeeping groups can be formed?
(b) If it is possible to have a timekeeping group with all the timekeepers being boys, what are the possible numbers of boys among the 16 students?
(c) If the probability of having a timekeeping group with all the timekeepers being boys is 3 / 364, find the number of boys among the 16 students.

(Answers:
(a) 160160;
(b) 10, 11, 12, 13, 14, 15, 16
(c) 12)

Homework Equations

Formulae for Bernoulli, Binomial, Geometric & Poisson Distributions

The Attempt at a Solution

I don't know how to solve part (c) of the question.

I tried:

_xC₁₀ / ₁₆C₁₀ = 3 / 364

and x can be found as 12.

I don't know if the above method is correct or not.

However, the question should be solved using the distribution formulae.

Another attempt:

x: number of boys within the 16 students

P(Boys) = x / 16

₁₆C₁₀ (x / 16)¹⁰ (1 - x / 16)⁶ = 3 / 364

But the x found is not correct.

Can anyone tell me how to solve it?

Thank you very much!

 

[CompuChip]

I think the second attempt is wrong, because you are using the binomial distribution there (why doesn't it apply?)

The first approach, although possible not worked out entirely correct, seems better. So let x be the number of boys in 16 students. You have to choose 10 students from the 16. What is the probability that they are all boys?