AND Probability Calculator (2024)

Head after head, that's the beginning of a lucky streak; but how do you define and calculate the probability of such events? Follow Omni in this tool dedicated to calculating the joint probability of two events.

What are you going to learn in this article?

• A short introduction to probability and a detailed explanation of what joint probability is. The conditions required and the formula we use when we learn how to find the probability of A and B, a generic pair of events.

• Examples and explanations of mathematical applications and real-world use of the joint probability formula. And much more!

To define the probability of multiple events, it's better to define what is probability first.

Probability is the likelihood of the outcome of a particular event. For example, the probability of getting a head in a coin toss is $P(\mathrm{head}) = 0.5$P(head)=0.5.

Where does this number come from? Super easy: toss the coin many, many times, and count the number of times you get a head, then divide this number by the number of launches. Say you throw it $100$100 times: we got head $52$52 times: $P(\mathrm{head}) = 0.52$P(head)=0.52. Try again! After another 100 flips you get $49$49 heads: $P(\mathrm{head}) = 0.49$P(head)=0.49.

Increase the number of tosses (though it will take time!): in a million launches, we counted $499607$499607 heads. This is our $0.5$0.5 probability.

When the number of outcomes is defined, we can also notice that the sum of the probabilities is $1$1. $P=1$P=1 is a particular probability: it represents an event we're sure will happen. The probability of obtaining either head or cross in a coin toss is $1$1 since either one of the events will happen. Au contraire, $P=0$P=0 represents an event that won't happen.

To calculate the joint probability of A and B, we need to impose an important condition: their independence. Two (or more) events are said to be independent if the outcome of each of them doesn't affect the others. For example:

• Throwing a die ten times represent ten separate, independent events; but

• Throwing ten dies at the same time in the same box represents events that influence each other: they are not independent.

The formula for the probability of A and B is straightforward. It's time to see it in action.

Say you want to calculate the probability of obtaining two $6$6's from two die tosses. That would be luck, right?

First, define the probability to get a $6$6 on the first die. Since there are six possible results, and we assume the die to not be tampered with, we say that:

It may happen that your events are not independent. In that case, the outcome of one event would affect the outcome of the other. If you want to calculate the joint probability of such events, you can't directly multiply the probability of the individual events, but consider a quantity called conditional probability.

The conditional probability of A given B, written $P(A|B)$P(A∣B), is the probability of the outcome A for the first event in case the other event had outcome B. We use this quantity to find $P(A\cap B)$P(A∩B) in the formula:

Probability and statistics are, we know, complex topics, often badly taught in school or directly overlooked. That's why Omni created a small suite of focused tools to help you with your problems and homework. Try our other probability calculators:

• The probability calculator; and
• The OR probability calculator.

To calculate the joint probability of two events, follow three easy steps:

1. Calculate the probability of the outcome A: P(A).

2. Calculate the probability of the outcome B: P(B).

3. Multiply these quantities together:
   P(A∩B) = P(A) × P(B)

The result is the probability that outcome A and outcome B will both happen.

The formula for the joint probability of independent events is simply a multiplication of the probability of the single events.

• For two events, A and B, the joint probability is:
  P(A∩B) = P(A) × P(B)

• For greater numbers of events, the generic formula is:
  P(A∩B∩C∩...) = P(A) × P(B) × P(C) × ...

The probability of obtaining two heads in two separate coin tosses is 0.25 or 25%.
To calculate this result:

• Define the probability of the two events. In our case:

  P(head) = 0.5

• Calculate the probability of two heads by multiplying the probability of two tosses:

  P(2 heads) = P(head) × P(head) = 0.5 × 0.5 = 0.25

Because the formula of the joint probability tells us that to find the result, we multiply together the probability of single events.

Since the probability of an event is always at most equal to 1, the multiplication of such values will necessarily cause the result to decrease. Look at the growing streak's probability:

• P(2 tails) = P(tail) × P(tail) = 0.5 × 0.5 = 0.25.

• P(3 tails) = 0.5 × 0.5 x 0.5 = 0.125.

• P(4 tails) = 0.5 × 0.5 x 0.5 x 0.5 = 0.0625.