probability

Question

A bag contains 7 red marbles, 4 blue marbles, 6 white marbles and 7 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

Answer 1

There are 7+4+6+7=24 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=(17/24)(16/23)(15/22)(14/21)(13/20)

XXXXY=(17/24)(16/23)(15/22)(14/21)(7/20)

XXXYX=(17/24)(16/23)(15/22)(7/21)(14/20)

XXYXX=(17/24)(16/23)(7/22)(15/21)(14/20)

XYXXX=(17/24)(7/23)(16/22)(15/21)(14/20)

YXXXX=(7/24)(17/23)(16/22)(15/21)(14/20)

Apart from no yellows (XXXXX), all the other probabilities are the same value, 119/1518, and their sum is 5(119/1518)=595/1518.

The probability for XXXXX (no yellows) is 221/1518. So (221+595)/1518=136/253=0.53755 approx or 53.755%.

The probability of less than two yellow marbles being drawn without replacement is about 53.76%.