The conditional probability that event A occurs, given that event B has occurred, is calculated as follows:
P(A|B) = P(A∩B) / P(B)
where:
- P(A∩B) = the probability that event A and event B both occur.Â
- P(B) = the probability that event B occurs.
Conditional probability is used in all types of areas in real life including weather forecasting, sports betting, sales forecasting, and more.
The following examples share how conditional probability is used in 4 real-life situations on a regular basis.
Example 1: Weather Forecasting
One of the most common real life examples of using conditional probability is weather forecasting.
Weather forecasters use conditional probability to predict the likelihood of future weather conditions, given current conditions.
For example, suppose the following two probabilities are known:
- P(cloudy) = 0.25
- P(rainy∩cloudy) = 0.15
A weather forecaster could use these values to calculate the probability that it will rain on a particular day, given that it is cloudy out:
- P(rain|cloudy) = P(rainy∩cloudy) / P(cloudy)
- P(rain|cloudy) = 0.15Â / 0.25
- P(rain|cloudy) = 0.6
The probability that it will rain given that it is cloudy out, is 0.6 or 60%.
This is a simplified example, but in real life weather forecasters use computer programs to take in data on current weather conditions and use conditional probability to calculate the likelihood of future weather conditions.
Example 2: Sports Betting
Conditional probability is frequently used by sports betting companies to determine the odds they should set for certain teams to win certain games.
For example, suppose the following two probabilities are known about some basketball team:
- P(Team A star player is hurt) = 0.15
- P(Team A wins∩Team A start player is hurt) = 0.02
The company could use these values to calculate the probability that team A will win, given that their star player is hurt:
- P(Team A Wins|star is hurt) = P(Team A Wins∩star is hurt) / P(star is hurt)
- P(Team A Wins|star is hurt) = 0.02 / 0.15
- P(Team A Wins|star is hurt) = 0.13
The probability that Team A will win given that their star player is hurt is is 0.13 or 13%.
If the sports betting company finds out ahead of the game that the star player is hurt, then they can use conditional probability to update their odds and payouts accordingly.
This happens all the time with sports betting companies when they calculate various odds for basketball, football, baseball, hockey matches, and more.
Example 3: Sales Forecasting
Retail companies use conditional probability to predict the chances that they’ll sell out of a certain product based on product promotions.
For example, suppose the following two probabilities are known:
- P(promotion) = 0.35
- P(sell out∩promotion) = 0.15
A retail company could use these values to calculate the probability that they’ll sell out of a certain product, given that a product promotion is ran that day:
- P(sell out|promotion) = P(sell out∩promotion) / P(promotion)
- P(sell out|promotion) = 0.15 / 0.35
- P(sell out|promotion) = 0.428
The probability that the retail company sells out of the product given that a promotion is ran that day is 0.428 or 42.8%.
If the retail company knows ahead of time that a promotion will be ran, they can increase their inventory ahead of time so they reduce the chances of selling out.
Example 4: Traffic
Traffic engineers use conditional probability to predict the likelihood of traffic jams based on stop light failures.
For example, suppose the following two probabilities are known:
- P(stop light failure) = 0.001
- P(traffic jam∩stop light failure) = 0.0004
A retail company could use these values to calculate the probability that they’ll sell out of a certain product, given that a product promotion is ran that day:
- P(traffic jam|stop light failure) = P(traffic jam∩stop light failure) / P(stop light failure)
- P(traffic jam|stop light failure) = 0.0004 / 0.001
- P(traffic jam|stop light failure) = 0.4
The probability that there will be a traffic jam given that there is a stop light failure is 0.4 or 40%.
Traffic engineers can use this conditional probability to decide if they need to design a different route to redirect traffic since a traffic jam is likely to occur if there is a traffic light failure.
Additional Resources
The following tutorials provide additional information about probability:
Probability vs. Proportion: What’s the Difference?
Probability vs. Likelihood: What’s the Difference?
Law of Total Probability: Definition & Examples