Demonstration of Role of Combinations in Probability

Let’s illustrate the power of counting techniques in the computation of probability with the following example. Here Combination is used as a counting technique. Combination is collection, here order is not important. In permutation order is important as permutation is arrangement.

Example: Four cards are drawn one after the other without replacement from a well shuffled deck of 52 cards. What is the probability of drawing three kings in these four draws?

Solution: Probability of a king in first draw = 4/52 [4 favorable cases and 52 total cases]

Probability of a king in second draw = 3/51 [3 favorable cases and 51 total cases]

Probability of a king in third draw = 2/50 [2 favorable cases and 50 total cases]

Probability of a non king card in the fourth draw=48/49[48 favorable cases and 49 total cases]

Probability of drawing three kings in these four draws =C(4,3) (4/52)(3/51)(2/50)(48/49)= 192/270725

Here order is not important, just receiving 3 Kings in a hand of 4 Cards counts.

Using Combinations - Probability of drawing three kings in these four draws = C(4, 3)C(48, 1)/C(52, 4) = 192/270725

If four cards are distributed to four players from a well shuffled deck of 52 cards, individual listing of possible combination of cards among four players becomes tedious and time consuming. Combination formula simplifies counting techniques of identifying set of favorable cases from set of total cases.

Industrial Application: A lot of 52 integrated chips contain 4 defective chips. While testing the quality of any lot of 52 integrated chips a sample of 4 chips is drawn. If more than two chips are found defective in the sample, the entire lot is rejected. What is the probability that this lot is rejected? Although there are only four defectives in the entire lot the entire lot is wrongly rejected. This is also called Producer’s risk!

P( 3 defectives) = C(4, 3)C(48, 1)/C(52, 4) = 0.00071

P(4 defectives) = C(4, 4)/C(52, 4)=1/27025

P( the lot is rejected) = 0.000714

This risk is 0.0714% only. So we can quantify uncertainty with probability.