What Is Conditional Probability? (Examples and Formulas)

By Indeed Editorial Team

Published 26 April 2022

The Indeed Editorial Team comprises a diverse and talented team of writers, researchers and subject matter experts equipped with Indeed's data and insights to deliver useful tips to help guide your career journey.

Many professionals use predictive models that can help them make more informed and intelligent decisions. Conditional probability forms the foundation of many prediction models and can assist companies in understanding their planning needs by helping them calculate the likelihood of different events. Understanding this term may help you prepare for analytical and strategic decision-making roles. In this article, we define this concept, provide formulas for calculating this probability and explain when to use this model.

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What is conditional probability?

Conditional probability defines the possibility that an event may happen based on other events. This method multiplies the probability of a preceding event with the probability of a succeeding or conditional event. The conditional event is an occurrence that has yet to happen and the event that you're trying to calculate the likelihood of occurring. This concept differs from unconditional probability because the latter type examines potential occurrences without examining other, similar, past events.

Events connected to a conditional event include any actions that might affect its outcome. For example, successfully getting into a prestigious university relies on a person's grades. The higher a person's grades are, the higher their chances of success are for getting admitted into a prestigious university. For example, someone that has an average grade of B- may have a 20% chance of getting into a prestigious university, while another person with an average grade of A+ has a 70% success chance.

Important probability terms and concepts

When calculating probabilities, there are various concepts, terms and formulas that can help improve accuracy. These include the:

  • Number of conditional factors: Multiple related factors may affect your probability outcome. For example, when drawing random tickets from a bag, the ticket number decreases with each draw and can affect the probability results by limiting selection options for subsequent draws.

  • Factor influence: Certain factors may have a larger impact than others on your probability. For example, grades may affect college acceptance more heavily than sports performance or volunteering.

  • Event type: Event type and occurrence may also heavily affect probability, such as if some events occur more frequently than others. Taking these event types into account helps make this process easier.

  • Other related events: Strong probability calculations consider all possible relationships and use multiple formulas when calculating these factors. As an event's complexity increases, the probability may become more accurate through multiple calculations that more precisely gauge an event's likelihood.

Calculating probabilities can help professionals in many careers. For example, insurance companies, business planners, politicians and engineers can use probability formulas when trying to predict outcomes from strategic decisions. Using this formula requires understanding a few simple mathematical concepts, such as multiplication and division, and how to gauge event types and their relationships.

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Understanding this probability type

The formula for calculating these probabilities is:

  • P(B|A) = P(A and B) / P(A)

The variables of the formula for calculating these probabilities are:

  • P: This term stands for an event's probability. You can display probability results as a decimal or a percentage, such as 53% or 0.53.

  • A and B: These terms show the conditional events connected in your formula and may include over two terms. You input the decimal percentage into this formula section and then follow the mathematical calculations outlined in the formula.

  • B|A: This term refers to the final probability after you finish calculating the formula.

How to calculate these probabilities

Use the steps below when calculating your probabilities:

1. Define your formula terms

Identify which terms you want to use for your probabilities formula. These include conditional events, such as job application acceptance rates, insurance offerings and potential career advancement. Choose the primary event, which refers to the most important event within your conditional formula.

2. Input terms into the formula

Add all your terms into the proper sections of the formula. Add as many values as you need to ensure the reliability of your calculation. Ensure the probability values of each event are in decimal form and not expressed as a percentage value.

3. Calculate the formula

Multiply all the terms within the P(A and B) section. Divide this result by the P(A) value to get your probability decimal and then multiply this result by 100 to get the percentage value.

Example: A job accepts 10% of all applicants and provides healthcare and insurance to a further 50% of those hired individuals, while 5% advance in their position. Here is the calculation to gauge the probability of all three results occurring:

  • P(B|A) = P (0.10 x 0.5 x 0.05)/P(0.10)

  • P(B|A) = 0.0025/0.10

  • P(B|A) = 0.025

In this example, the probability of getting hired by this company, receiving health insurance and advancing in the position is 2.5% or 0.025.

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Ways to use these probabilities

You may use these probabilities when performing multiple statistical analysis processes. It helps companies and individuals better understand the success rates of decisions and projects. A few ways that you can use these probabilities include:

Calculating important business decisions

Businesses may use this probability when assessing different situations. For example, a company's internal training results in 45% passing rates for first aid, 67% passing rates for on-the-job training and a total passing percentage of 35% when combined with a test for executive positions, which you can input into the formula:

P(B|A) = P (0.45 x 0.67 x 0.35)/P(0.45) = 0.23 * 100% = 23%

This company's overall internal training passing rate is 23%. The company can use this information to identify its training's strengths and weaknesses to identify areas for improvement and track its progress.

Gauging potential investment changes

Investors can use these probabilities when gauging multiple unique investment opportunities. For example, they find one stock increases in price 35% of the time each first quarter, 15% of the time in the second, 5% of the time in the third and a total of 68% for the year. Multiply these numbers together and divide by 0.35 to get 0.05 or a 5% potential for an increase in the fourth quarter. These probabilities help investors predict different possibilities and combine the results with other information, such as industry research, to make more informed investment decisions.

Understanding unpredictable events

Many professionals use these probabilities when planning for any unforeseen events. For example, a survey finds rainstorms occur on 5% of all days in June, 10% on weekends and 1% on odd-numbered days. Multiply these three numbers together and divide by 0.05 to get 0.001 or 0.1% chance of rainstorms occurring in June and on an odd-numbered day. These statistical models can help meteorologists build long-term weather forecasts. Statisticians may also expand on these formulas to include more advanced calculations, such as predicting the chance of hurricane winds during rainstorms.

FAQs about probabilities

Here are the answers to some frequently asked questions about different probability models:

What is prior probability?

The prior probability concept examines the likelihood of an event before you gather any information or do any research. For example, you may use a prior belief about stock behaviour when choosing a potential investment option. Research professionals may use this concept when assessing the effect of bias on people's assumptions by comparing prior probabilities to more carefully researched and calculated probabilities.

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What is compound probability?

Compound probability examines two independent but identical events and then examines their related possibility. For example, flipping a coin to a specific side has a 50% probability and when multiplied by the 50% probability of a second flip, you get a 25% chance of flipping the same side again. Statisticians and business professionals can use this probability model when calculating the probability of multiple occurrences of a single event.

What is joint probability?

Joint probability measures the likelihood of two events with different probabilities occurring and calculates the intersection of two or more of these events. For example, picking a heart from a deck of cards has a 25% probability, while picking a specific card value has 7.69%. When calculating the probability of picking a specific card, like a king of hearts, you can multiply these two results to get a 1.9% probability. This type of probability model works well when calculating the probability of related but not identical events occurring.

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