Probability Examples on Combinations

In this class, We discuss Probability Examples on Combinations.

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The reader should have prior knowledge of mutually exclusive events. Click Here.

Example 1:

A bag contains eight black balls and six green balls. Find the probability of drawing two balls of the same color.

The example is dependent on the permutations and combinations.

If the reader is not good at permutations and combinations, please check our course aptitude for placement and competitive exams.

We refresh the concept of combinations once and move into the probability.

We have eight black balls.

We numbered the balls from one to eight.

Does the below diagram show How many ways we can select two black balls?

In total, we have 28 different combinations.

We obtain the value 28 by using 8c2.

8c2 = (8*7)/(2*1) = 28

In the bag, we have eight black balls and six green balls.

Total fourteen balls.

Sample space S = 14 c2.

We can select two balls from the bag in 14 c2 ways.

Event E1 = {Two black balls}.

We can select two black balls in 8c2 ways.

Event E2 = {Six green balls}

We can select six green balls in 6c2 ways.

The two events, E1 and E2, are mutually exclusive.

P(E1) = (8c2)/(14c2)

P(E2) = (6c2)/(14c2)

P(E1 U E2) = P(E1) + P(E2)

P(E1 U E2) = (8c2)/(14c2) + (6c2)/(14c2)

Example 2:

A box contains six green, four white, and five black balls. Randomly pick four balls from the bag.

Find the probability that there is atleast one ball of each color among the balls drawn.

Solution:

Total, we have fifteen balls.

Below are the chances of occurring the event with atleast one ball of each color.

1) 1 green, 1 white, 2 black = E1

2) 1 green, 2 white, 1 black = E2

3) 2 green, 1 white, 1 black = E3

Sample space S = 15c4

The events E1 and E2, and E3 are mutually exclusive.

P(E1 U E2 U E3) = P(E1) + P(E2) + P(E3)

P(E1) = (6c1 * 4c1 * 5c2) /15c4

P(E2) = (6c1 * 4c2 * 5c1) /15c4

P(E3) = (6c2 * 4c1 * 5c1) /15c4

P(E1 U E2 U E3) = (6c1 * 4c1 * 5c2) /15c4 + (6c1 * 4c2 * 5c1) /15c4 + (6c2 * 4c1 * 5c1) /15c4