Conditional Probabilities: Learn It 2

  • Describe and find conditional probabilities.
  • Understand the concept of independent events.

Conditional Probability

A conditional probability is calculated based on the assumption that one event has already occurred. Conditional probabilities restrict the total. The new total is indicated after the word “given” in the question.

conditional probability

The conditional probability of [latex]A[/latex] given [latex]B[/latex] is written [latex]P(A\text{ given }B)[/latex] or [latex]P(A|B)[/latex].

 

[latex]P(A\text{ given }B)[/latex] is the probability that event [latex]A[/latex] will occur given that the event [latex]B[/latex] has already occurred.

 

[latex]P(A|B)=\frac{P(A \text{ and }B)}{P(B)}[/latex]

To calculate any probabilities, a contingency table is typically used to provide a way of portraying data that can facilitate in calculating probabilities. The table helps in determining conditional probabilities quite easily as well.

A researcher conducts a survey of 120 randomly selected college students to try to answer the questions: If someone has a laptop, are they likely to own a desktop computer? If someone has a desktop computer, are they likely to own a laptop? The results of the survey are displayed in the following contingency table.

  Owns laptop Does not own laptop Total
Owns desktop 20 20 40
Does not own a desktop 60 20 80
Total 80 40 120

If someone does not own a desktop, what is the probability that they don’t own a laptop computer either?

Appropriate alternative text can be found in the description above.
Figure 1. This two-way table displays computer ownership among 120 people, showing how many own a laptop, a desktop, both, or neither.