Independent Events and Multiplication Rules

Topic Content:

Two events, ‘A’ and ‘B,’ are said to be independent if the occurrence or non-occurrence of event ‘A’ does not affect the occurrence of event ‘B’.

Mathematically;

Pr(A and B) = Pr(A) × Pr(B)

Pr(A ∩ B) = Pr(A) × Pr(B)

When a probability of event ‘A’ affects the probability of event ‘B’, we say the events are dependent, and the probability is said to be conditional .

Example 7.2.1:

A box contains 5 blue balls, 3 black balls and 2 red balls of the same size. A ball is selected at random from the box and replaced. Find the probability of obtaining:

(a) two red balls
(b) two blue or two black balls
(c) one black ball and one red ball in any order. (WAEC)

Solution

Note: when replacement is done, it is an independent event but when not replaced, it is a dependent event.

Total balls = 5 + 3 + 2 = 10

(a) P r(two red) = \( \frac{2}{10} \: \times \frac{2}{10} \\ = \frac{4}{100} \\ = \frac{1}{25} \)

(b) Pr (two blue or two black) = B.B + BL.BL

= \( \left( \frac{5}{10} \: \times \frac{5}{10} \right) \: +\: \left( \frac{3}{10} \: \times \frac{3}{10} \right) \\ = \frac{25}{100} \: + \: \frac{9}{100} \\ = \frac{34}{100} \\ = \frac{17}{50}\)

(c) Pr (one black and one red) = BL.R + R.BL

= \( \left( \frac{3}{10} \: \times \frac{2}{10} \right) \: +\: \left( \frac{2}{10} \: \times \frac{3}{10} \right) \\ = \frac{6}{100} \: + \: \frac{6}{100} \\ = \frac{12}{100} \\ = \frac{3}{25}\)

Example 7.2.2:

A building contractor tendered