How to Identify Independent and Dependent Events?

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A step-by-step guide to independent and dependent events

There are two types of events in probability, often classified as independent events or dependent events:

Independent events:

If the outcome of one event does not affect the outcome of another event, two events are called independent events. Or we can say that if one event does not affect the probability of another event, it is called an independent event.

Independent events formula:

If two events \(A\) and \(B\) are independent, then the probability of happening of both \(A\) and \(B\) is:

\(\color{blue}{P\left(A\:⋂\:B\right)=P\left(A\right).\:P\left(B\right)}\)

Dependent events:

If the outcome of one event affects the outcome of another event, two events are said to be dependent. More likely, dependent events are usually actual events that rely on another event to occur.

Dependent events formula:

If \(A\) and \(B\) are dependent events, then the probability of \(A\) and \(B\) occurring is:

\(\color{blue}{P\left(B\:and\:A\right)=P\left(A\right)×P\:\left(B\:after\:A\right)}\)

How to identify independent events?

Before applying probability formulas, an independent or dependent event must be identified. A few steps to check if the probability belongs to a dependent or independent event:

• Step 1: Check if it is possible for the events to happen in order. If yes, go to step \(2\) or go to step \(3\).
• Step 2: Check if one event affects the outcome of the other event. If yes, go to step \(4\), or else go to step \(3\).
• Step 3: The event is independent. Use the formula of independent events and get the answer.
• Step 4: The event is dependent. Use the formula of the dependent event and get the answer.

Independent and Dependent Events – Example 1:

A juggler has seven red, five black, and four yellow balls. During the stunt, he accidentally drops a ball and doesn’t pick it up. As he continues, another ball drops. What is the probability that the first ball dropped is yellow and the second ball is black?

Solution:

The probability of the first ball is yellow: \(P\) \(=\frac{4}{16}\) 

The probability of the second ball is black: \(P\)\(= \frac {5}{15}\)

\(P\left(yellow\:than\:black\right)\:=\:P\left(yellow\right)×\:P\left(black\right)\:\)
\(=\frac{4}{16}× \frac{5}{15}=\frac{1}{12}\)