Probability

Question

A bag contains 7 red marbles, 7 blue marbles, 11 white marbles and 5 yellow marbles. You are asked to draw 5 marbles from the bag without replacement. What is the probability of drawing less than two yellow marbles?

Answer 1

There are 7+7+11+5=30 marbles in the bag. Less than two implies one or no yellow marbles to be drawn.

This is another way of saying that at least 4 marbles are not to be yellow.

Let X represent any other colour but yellow, while Y represents yellow.

The sequence possibilities are:

XXXXX XXXXY XXXYX XXYXX XYXXX YXXXX.

These 6 possible outcomes have corresponding combined 1probabilities:

XXXXX=(25/30)(24/29)(23/28)(22/27)(21/26)

XXXXY=(25/30)(24/29)(23/28)(22/27)(5/26)

XXXYX=(25/30)(24/29)(23/28)(5/27)(22/26)

XXYXX=(25/30)(24/29)(5/28)(23/27)(22/26)

XYXXX=(25/30)(5/29)(24/28)(23/27)(22/26)

YXXXX=(5/30)(25/29)(24/28)(23/27)(22/26)

Apart from no yellows (XXXXX), all the other probabilities are the same value, 6325/71253, and their sum is 5(6325/71253)=31625/71253.

The probability for XXXXX (no yellows) is 1265/3393. So 31625/71253+1265/3393=58190/71253=0.81667 or 81.67% approx.

The probability of less than two yellow marbles being drawn without replacement is about 81.67%.