7.2 Probability of the Combination of Two Events

7.2 Probability of the Combination of Two Events

1. For two events, A and B, in a sample space S, the events A ∩ B (A and B) and A υ B (A or B) are known as combined events.

2. The probability of the union of sets A and B is given by:

P (A \cup B) = P (A) + P (B) - P (A \cap B)

3. The probability of the union of sets A and B can also be calculated using an alternative method, i.e.

P (A \cup B) = \frac{n (A \cup B)}{n (S)}

4. The probability of event A and event B occurring, P(A ∩ B) can be determined by the following formula.

P (A \cap B) = \frac{n (A \cap B)}{n (S)}

Example:

Given a universal set ξ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. A number is chosen at random from the set ξ . Find the probability that

(a) an even number is chosen.

(b) an odd number or a prime number is chosen.

Solution:

The sample space, S = ξ

n(S) = 14

(a)

Let A = Event of an even number is chosen

A = {2, 4, 6, 8, 10, 12, 14}

n (A) = 7

\begin{array}{l} P (A) = \frac{n (A)}{n (S)} \\ = \frac{7}{14} = \frac{1}{2} \end{array}

(b)
Let,

B = Event of an odd number is chosen

C = Event of a prime number is chosen

B = {3, 5, 7, 9, 11, 13, 15} and n (B) = 7

C = {2, 3, 5, 7, 11, 13} and n (C) = 6

The event when an odd number or a prime number is chosen is B υ C.

P (B υ C) = P (B) + P (C) – P (B ∩ C)

B ∩ C = {3, 5, 7, 11, 13}, n (B ∩ C) = 5

\begin{array}{l} P (B \cup C) \\ = P (B) + P (C) - P (B \cap C) \\ = \frac{n (B)}{n (S)} + \frac{n (C)}{n (S)} - \frac{n (B \cap C)}{n (S)} \\ = \frac{7}{14} + \frac{6}{14} - \frac{5}{14} \\ = \frac{8}{14} = \frac{4}{7} \\ ∴ The probability of choosing an \\ odd number or a prime number = \frac{4}{7} . \end{array}