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 the previous section we computed the probabilities of events that were independent of each other. We saw that getting a certain outcome from rolling a die had no influence on the outcome from flipping a coin, even though we were computing a probability based on doing them at the same time.
In this section, we will consider events that are dependent on each other, 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”
For example, if you draw a card from a deck, then the sample space for the next card drawn has changed, because you are now working with a deck of 51 cards. In the following example we will show you how the computations for events like this are different from the computations we did in the last section.
example
What is the probability that two cards drawn at random from a deck of playing cards will both be aces?
Conditional Probability Formula
If Events A and B are not independent, then
P(A and B) = P(A) · P(B | A)
Converting a fraction to decimal form
Probabilities can be expressed in fraction or decimal form. To convert a fraction to a decimal, use a calculator to divide the numerator by the denominator.
Ex. [latex]\dfrac{19}{51}=19 \div 51 \approx 0.3725[/latex]
example
If you pull 2 cards out of a deck, what is the probability that both are spades?
Try It
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:
- has a speeding ticket given they have a red car
- 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
If you draw two cards from a deck, what is the probability that you will get the Ace of Diamonds and a black card?
These two playing card scenarios are discussed further in the following video.
Try It
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
- P(not pregnant | positive test result)
- 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.
Try It
Candela Citations
- Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
- Math in Society. Authored by: Lippman, David. Located at: http://www.opentextbookstore.com/mathinsociety/. License: CC BY-SA: Attribution-ShareAlike
- Statistics, Describing Data, and Probability . Authored by: Jeff Eldridge. License: CC BY-SA: Attribution-ShareAlike
- Question ID 7118. Authored by: WebWork-Rochester. License: CC BY: Attribution. License Terms: IMathAS Community License CC-BY + GPL