Conditional Probability

Learning Outcomes

  • Calculate a conditional probability using standard notation

LEARNING PROBABILISTIC PROCESSES

Remember to work through each example in the text and in the EXAMPLE and TRY IT boxes with a pencil on paper, pausing as frequently as needed to digest the process. Watch the videos by working them out on paper, pausing the video as frequently as you need to make sense of the demonstration. Don’t be afraid to ask for help — hard work and willingness to learn translate into success!

In this section, we will consider events where additional information is given. Conditions on an event means you are considering a new, smaller set of outcomes where that event has already occurred. The probability of such events is called conditional probabilities.

Conditional Probability

The probability the event B occurs, given that event A has happened, is represented as

P(B | A)

This is read as “the probability of B given A”

Example

The table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:

  1. has a speeding ticket given they have a red car
  2. has a red car given they have a speeding ticket
Speeding ticket No speeding ticket Total
Red car 15 135 150
Not red car 45 470 515
Total 60 605 665

These kinds of conditional probabilities are what insurance companies use to determine your insurance rates. They look at the conditional probability of you having accident, given your age, your car, your car color, your driving history, etc., and price your policy based on that likelihood.

View more about conditional probability in the following video.

Example

A home pregnancy test was given to women, then pregnancy was verified through blood tests.  The following table shows the home pregnancy test results.

Find

  1. P(not pregnant | positive test result)
  2. P(positive test result | not pregnant)
Positive test Negative test Total
Pregnant 70 4 74
Not Pregnant 5 14 19
Total 75 18 93

See more about this example here.

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