Dependent Events and Independent Events

9.2

Dependent Events and Independent Events

Differentiate Between Dependent and Independent Events

Combined events can be categorised as dependent and independent events.
Event \(A\) and event \(B\) are independent events if the occurrence of event \(A\) has no effect on the occurrence of event \(B\) and vice versa.
In other words, event \(A\) and event \(B\) are dependent events if the occurrence of event \(A\) affects the occurrence of event \(B\).

Example

Identify whether the following combined events are dependent events or independent events. Justify your answers.

1. Taking an Uber ride and getting a free meal at your favourite restaurant.

Solution:

It is an independent event because the probability of taking an Uber does not affect the probability of getting a free meal at favourite restaurant.

2. Getting into a traffic accident is dependent upon driving or riding in a vehicle.

Solution:

It is a dependent event because the probability of getting into a traffic accident has affect driving or riding in a vehicle

Verify Conjecture About The Formula of Probability of Combined Events

The probability of the intersection of two intersection events \(A\) and \(B\) is equal to the product of the probability of event \(A\) and probability of event \(B\).

Formula

\(P(A \space \text{and} \space B) = P(A) \times P(B)\)

Example

Khairil has \(40\) cards consisting of white, blue and red. If a card is chosen at random, the probability of choosing a red card is \(\dfrac{3}{5}\). Calculate

(a) number of cards in red.

(b) the probability of choosing a blue card if Khairil has \(8\) white cards.

Solution: If

\(P\)	White card event selected.
\(B\)	Blue card event selected.
\(M\)	Red card event selected.
\(S\)	Sample space.

(a) Number of cards in red.

\(\begin{aligned}n(S) &= 40\\ n(M)&=P(M)\times n(S)\\&=\dfrac{3}{5}\times40\\&=24\end{aligned}\)

(b) The probability of choosing a blue card if Khairil has \(8\) white cards.

\(\begin{aligned}n(P) &= 8\\ n(B) &= 40 – 24 – 8 = 8\\\,P(B)&=\dfrac{n(B)}{n(S)}\\&=\dfrac{8}{40}\\&=\dfrac{1}{5} \end{aligned}\)