Question

bag contains 7 red marbles, 5 white marbles, and 8 blue marbles. You draw 5 marbles out at random, without replacement. What is the probability that all the marbles are red?

The probability that all the marbles are red is  .

What is the probability that exactly two of the marbles are red?


What is the probability that none of the marbles are red?

Answer 1

Solution

Back-up Theory

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}….................................................…(1)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN(Number, Number_chosen) [Number is n, Number_chosen is r]……………………………………............................................................................…. (1a)

Probability of an event E, denoted by P(E) = n/N …………………........................................……………………..…………(2)

where n = n(E) = Number of outcomes/cases/possibilities favourable to the event E and N = n(S) = Total number all possible outcomes/cases/possibilities.

Now to work out the solution,

In total, there are 20 marbles.

Drawing 5 marbles out of 20 at random, without replacement, the number of possibilities is: [vide (1)]

²⁰C₅ = 15504. [vide (1a)]

So, vide (2), N = 15504 …………………………………………………………..........................................………………….. (3)

Part (A)

If all 5 marbles are to be red, these 5 must come from 7 red marbles available. This, vide (1), can be selected in ⁷C₅ = 21 ways. Thus, vide (2), n = 21 and hence

The probability that all the marbles are red is: 21/15504 = 0.0014 Answer

Part (B)

If exactly 2 marbles are to be red, these 2 must come from 7 red marbles and the remaining 3 must come from the remaining 13 marbles available. This, vide (1), can be selected in (⁷C₂)(¹³C₃)= (21 x 286) = 6006 ways. Thus, vide (2), n = 6006 and hence

The probability that exactly 2 marbles are red is: 6006/15504 = 0.3873 Answer

Part (C)

If none of 5 marbles is to be red, the 5 marbles must come from the remaining 13 non-red marbles available. This, vide (1), can be selected in ¹³C₅ = 1287 ways. Thus, vide (2), n = 1287 and hence

The probability that none of the marbles is red is: 1287/15504 = 0.0830 Answer

DONE