A Joint Probability: What Is It?
In probability theory, the likelihood that two occurrences will occur simultaneously is known as a joint probability. Put differently, joint probability represents the chance of two events happening simultaneously.
Where:
- P(A ⋂ B) is the notation for the joint probability of event “A” and “B”.
- P(A) is the probability of event “A” occurring.
- P(B) is the probability of event “B” occurring.
Joint Probability and Independence
The events have to be independent for joint probability calculations to function properly. Stated otherwise, there must be no mutual interaction between the occurrences. It’s crucial to consider whether the result of one event might affect the result of the other in order to establish whether two occurrences are independent or dependent on one another. The occurrences are independent if the result of one does not influence the result of the other.
The likelihood of clouds in the sky and the likelihood of rain on that particular day are two examples of dependent occurrences. The likelihood of rain on a given day depends on the likelihood of clouds in the sky. As a result, they are interdependent events.
An illustration of The likelihood of two coin flips ending in heads is known as an independent event. The likelihood of receiving heads in the second coin toss is independent of the first coin toss’s probability of producing heads.
Examples
The following are examples of joint probability:
Example 1
What is the joint probability of rolling the number five twice in a fair six-sided dice?
Event “A” = The probability of rolling a 5 in the first roll is 1/6 = 0.1666.
Event “B” = The probability of rolling a 5 in the second roll is 1/6 = 0.1666.
Therefore, the joint probability of event “A” and “B” is P(1/6) x P(1/6) = 0.02777 = 2.8%.
Example 2
What is the joint probability of drawing a number ten card that is black?
Event “A” = The probability of drawing a 10 = 4/52 = 0.0769
Event “B” = The probability of drawing a black card = 26/52 = 0.50
Therefore, the joint probability of event “A” and “B” is P(4/52) x P(26/52) = 0.0385 = 3.9%.
FAQ’s
What’s the difference between regular probability and joint probability?
Regular probability tells you the chance of a single event happening, like flipping heads on a coin. Joint probability, on the other hand, is like asking about two things at once. It’s the chance of both events happening together, like getting heads on two separate coin flips.
Do the events have to be best friends for joint probability to work?
Not necessarily! But they do need to be independent. This means one event happening shouldn’t affect the chance of the other happening. Imagine rolling two dice. Getting a six on the first die doesn’t change the chance of getting a three on the second. That’s independence! But if you’re looking at the weather, things get trickier. Rain is more likely if it’s already cloudy. So, weather events are often dependent, and joint probability wouldn’t be the best tool for them.
Sounds complicated, can you give an example?
Sure! Let’s say you have a bag with 3 red marbles and 3 blue marbles. You pull out a marble, don’t put it back, and then pull out another. What’s the chance of getting red, then blue? For the first marble, there’s a 3 in 6 chance (half) of getting red. But since you didn’t put it back, there are only 5 marbles left. Now, there are only 3 blue marbles left out of those 5. So, the chance of getting blue after a red marble is 3 out of 5. This means the joint probability is (3/6) x (3/5) = 0.3 or 30%.
Is there a fancy way to write joint probability?
Absolutely! Mathematicians use the symbol P(A ∩ B) to represent the joint probability of event A happening and event B happening. The ∩ symbol kind of looks like two circles overlapping, showing that we’re considering both events together.
