What is a 'Joint Probability'

A joint probability is a statistical measure that calculates the likelihood of two events occurring together and at the same point in time. Joint probability is the probability of event Y occurring at the same time event X occurs.

Notation for joint probability takes the form:

P(X ∩ Y) or P(X and Y) or P(XY), which reads as the joint probability of X and Y.

BREAKING DOWN 'Joint Probability'

Probability is a field of statistics that deals with the likelihood of an event or phenomena occurring. It is quantified as a number between 0 and 1 inclusive, where 0 indicates an impossible chance of occurrence and 1 denotes the certain outcome of an event. For example, the probability of drawing a red card from a deck of cards is ½ = 0.5. This means that there is an equal chance of drawing a red and drawing a black; since there are 52 cards in a deck, of which 26 are red and 26 are black, there is a 50-50 probability of drawing a red card vs. a black card.

Joint probability is a measure of two events happening at the same time, and can only be applied to situations where more than one observation can occur at the same time. For example, from a deck of 52 cards, the joint probability of picking up a card that is both red and 6 is P(6 ∩ red) = 2/52 = 1/26, since a deck of cards has two red sixes — the six of hearts and the six of diamonds. You can also use a formula to calculate the joint probability — P(6 ∩ red) = P(6) x P(red) = 4/52 x 26/52 = 1/26.

The symbol “∩” in a joint probability is referred to as an intersection. The probability of event X and event Y happening is the same thing as the point where X and Y intersect. Therefore, joint probability is also called the intersection of two or more events. A Venn diagram is perhaps the best visual tool to explain an intersection.

A Venn diagram of the joint probability of drawing a red card that and a 6

From the Venn above, the point where both circles overlap is the intersection, which has two observations — the six of hearts and the six of diamonds.

Joint Probability vs. Conditional Probability

Joint probability should not be confused with conditional probability — the probability that one event will happen given that another action or event happens. The conditional probability form is P(X, given Y) or P(X│Y), which is to say, the chance of one event happening is conditional on another event happening. For example, from a deck of cards, the probability that you get a six, given that you drew a red card is P(6│red) = 2/26 = 1/13, since there are two sixes out of 26 red cards.

Joint probability only factors the likelihood of both events occurring. Conditional probability can be used to calculate joint probability, as seen in this formula: P(X ∩ Y) = P(X│Y) x P(Y). The probability that A and B occurs is the probability of X occurring, given that Y occurs multiplied by the probability that Y occurs. Given this formula, the probability of drawing a 6 and a red at the same time will be P(6 ∩ red) = P(6│red) x P(red) = 1/13 x 26/52 = 1/13 x 1/2 = 1/26.

Statisticians and analysts use joint probability as a tool when two or more observable events can occur simultaneously. For instance, joint probability can be used to estimate the likelihood of a drop in the Dow Jones Industrial Average (DJIA) accompanied by a drop in Microsoft’s share price, or the chance that the value of oil rises at the same time the U.S. dollar weakens.

  1. Compound Probability

    Compound probability is a mathematical term relating to the likeliness ...
  2. Conditional Probability

    Conditional probability is the probability of an event or outcome ...
  3. Posterior Probability

    Posterior probability is the revised probability of an event ...
  4. Probability Distribution

    A probability distribution is a statistical function that describes ...
  5. Yearly Probability Of Living

    Yearly probability of living is a numerical figure that depicts ...
  6. Gas (Ethereum)

    Gas is the pricing value required to conduct a transaction or ...
Related Articles
  1. Investing

    Financial Forecasting: The Bayesian Method

    This method can help refine probability estimates using an intuitive process.
  2. Investing

    Scenario Analysis Provides Glimpse Of Portfolio Potential

    This statistical method estimates how far a stock might fall in a worst-case scenario.
  3. Investing

    How This New ETF Tracks Millennial Consumer Habits

    A recently launched ETF aims to track Millennial consumer habits.
  4. Financial Advisor

    10 Companies With No Debt (DOX,NHTC,PAYX)

    These 10 companies have no debt, a big positive in today's economic environment. Three stand out above the rest.
  5. Managing Wealth

    6 Reasons Why You Might Not Elect Joint Ownership

    Joint ownership offers numerous benefits, but there are underlying issues to take into account.
  6. Insights

    Target Red Card: What You Need to Know (TGT)

    Target Corp’s Red Card is fantastically popular and a win-win for shoppers and the retailer.
  7. Trading

    The Math Behind Betting Odds & Gambling

    A betting odd opportunity should be considered valuable if the probability assessed for an outcome is higher than the implied probability estimated by the bookmaker. Read more on the math behind ...
  8. IPF - Banking

    Should You Open A Joint Bank Account?

    Joint finances may not be romantic, but it's an issue that serious long-term couples will have to confront at some point.
  9. Investing

    What Are The Odds Of Scoring A Winning Trade?

    Just because you're on a winning streak doesn't mean you're a skilled trader. Find out why.
  10. Investing

    The 10 Largest Holdings in the S&P 500 (AAPL,AMZN)

    What is the investing potential of each of the 10 largest companies in the S&P 500 Index?
  1. How can you find the demand function from the utility function?

    Learn how the utility function can be used to derive the demand function, and how both of these concepts relate to utility ... Read Answer >>
  2. What are the primary disadvantages of forming a joint venture?

    Learn the disadvantages to forming and maintaining a joint venture partnership, including factors business owners should ... Read Answer >>
  3. What are the primary advantages of forming a joint venture?

    Understand what the advantages of a joint venture are and discover what make this business strategy a good alternative to ... Read Answer >>
Trading Center