Imagine you are putting together musical performances and yo

Imagine you are putting together musical performances and you are employing musicians
at random. There are many available for hire: 23 guitarists, 15 vocalists, 6 drummers, 14
bassists, 8 violinists, 9 violas, 6 cellists.

a)We now move to a different city and the musicians for hire are different. Here,
we have 10 guitarists, 10 vocalists, 10 drummers, 10 bassists. What is the probability
we form a rock band when hiring four musicians at random? In detail.

b)Given the same situation in part (a), what is the probability we get two pairs
of musicians (e.g. two guitarists and two bassists or two drummers and two bassists)? In detail.

c)Given the same situation in part (a), what is the probability we get all four
musicians be the same type? In detail

Solution

I am assuming that you want to use the data given in part (A) and not the one above in the main question body.

a)

The ways in which I can hire 4 musicians = ⁴⁰C₄

To form a rock band, we need to have one of each (assumption) :

So, number of ways to do that = ¹⁰C₁ * ¹⁰C₁ *¹⁰C₁* ¹⁰C₁

Thus, the required probability = ¹⁰C₁ * ¹⁰C₁ *¹⁰C₁* ¹⁰C₁ /  ⁴⁰C₄ = 1000 / 9139 = 0.1094

b)

For getting two pairs, we need to choose two from the set of tens twice. Also, there must be choice of their specialization (i.e guitarist, vocalist, drummers, bassists)

Thus, number of ways =  (⁴C₂ ) * [ ¹⁰C₂ * ¹⁰C₂ ]

Probability =  (⁴C₂ ) * [ ¹⁰C₂ * ¹⁰C₂ ] / ⁴⁰C₄ = 1215 / 9139 = 0.1329

c)

Over here we proceed just as above i.e choose 4 out of 10 and also choose one of the specializations out of 4

Thus, number of ways =  (⁴C₁ ) * [ ¹⁰C₄ ]

Probability =  (⁴C₁ ) * [ ¹⁰C₄ ] / ⁴⁰C₄ = 84 / 9139 = 0.00919

Hope this helps.Ask if yo have doubts