How to Identify Independent and Dependent Events?
Two events are independent when the outcome of one does not change the probability of the other, and dependent when it does. The practical test is whether the first event removes something from the pool. The examples below apply that test, with practice problems and answers.
Identify Independent and Dependent Events: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- List outcomesName the possible results or count them carefully.
- Choose the ruleUse addition, multiplication, or conditional probability based on the wording.
- Check the rangeA probability must be between 0 and 1.
Worked examples
Simple probability
- Even outcomes are 2, 4, and 6.
- There are 3 favorable outcomes out of 6.
- Reduce the fraction.
Independent events
- P(heads) = 1/2 for each flip.
- The flips are independent.
- Multiply 1/2 by 1/2.
Try one before moving on
Identify Independent and Dependent Events: pop-up practice
Related Topics
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, the 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}{Pleft(A:⋂:Bright)=Pleft(Aright).:Pleft(Bright)})
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}{Pleft(B:and:Aright)=Pleft(Aright)×P:left(B:after:Aright)})
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 the events can 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})
(Pleft(yellow:than:blackright):=:Pleft(yellowright)×:Pleft(blackright):)
(=frac{4}{16}× frac{5}{15}=frac{1}{12})
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