Probability Calculator

The Probability Calculator is a free online tool that calculates the probability of a single event based on the provided number of events. It can also determine the probabilities of independent events for the following eight scenarios:

• A and B both occurring: P(A ∩ B)
• A or B both occurring: P(A ∪ B)
• A Not occurring: P(A')
• B Not occurring: P(B')
• A or B occurs but Not both: P(A Δ B)
• Neither A nor B occurring: P((A ∪ B)')
• A occurring but not B: P(A but NOT B)
• B occurring but not A: P (B but NOT A).

An option is also available to see all of the above as a whole.

What is Probability?

Probability is the measure of the likelihood or chance of an event happening. Probability values range from 0 to 1; where 0 denotes the event is impossible and 1 shows the event is certain. It quantifies the uncertainty of events and makes predictions about their occurrence.

Probability can be written in percentages, fractions, or decimals. For example, the probability of rolling a 6 on a standard die is 1/6 which is equivalent to 16.67% or 0.1667.

Probability Formulas

Probability has two types of formulas, one for the single event and the other for the independent event.

The formula for Single Event Probability

Single-event probability is the likelihood of a specific outcome occurring in a single trial.

1. Probability of an Event (E) Formula

The formula for finding the probability of a single event (E) is given below.

Probability of Event E = Number of Events Occurred / Number of Possible Outcomes

P(E) = n(E) / n(T)

To calculate the probability P(E) as a percentage, multiply the probability P(E) by 100.

P(E)% = P(E) x 100

2. Complement Rule

Formula to calculate the percentage of the complement of the event E.

P(E’)% = [1 – P(E)] x 100

Probability Formulas for Independent Events

Two events are considered independent when the occurrence of one event does not affect the probability of another event happening. Use the following formulas to find the probability of independent events.

1. A and B both occurring: P(A ∩ B)

P(A ∩ B) = P(A) × P(B)

2. A or B both occurring: P(A ∪ B)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

3. A Not occurring: P(A')

P(A') = 1 - P(A)

4. B Not occurring: P(B')

P(B') = 1 - P(B)

5. A or B occurs but Not both: P(A Δ B)

P(A Δ B)=P(A)+P(B)−2×P(A∩B)

6. Neither A nor B occurring: P((A ∪ B)')

P((A ∪ B)') = 1 - P(A ∪ B)

7. A occurring but not B: P(A but NOT B)

P(A but NOT B) = P(A) - P(A ∩ B)

8. B occurring but Not A: P(B but NOT A).

P(B but NOT A) = P(B) - P(A ∩ B)

How to Calculate Probability?

Let’s consider some examples to learn how to find the probability of single events and independent events.

Example 1: (For Single Event Probability)

A fair die is rolled. What is the probability that it shows the even numbers?

Solution:

Step 1: Determine the number of the Possible Outcomes.

A fair six-sided die has six faces so n(T) = 6

Step 2: Determine the number of events that occurred n(E)

The even numbers on a single die are 2, 4, and 6.

n(E) = 3

Step 3: Apply the Probability Formula.

P(E) = n(E) / n(T)

Step 4: Substitute Values.

­P(E) = 3/6

­P(E) = 0.5 or 50%

So the probability of rolling an even number on the fair six-sided die is 0.5 or 50%.

Example 2: (For Independent Event Probability)

Two students A and B can solve 60% and 75% of the problems from the exercise, respectively. Find the probability that either A or B can solve a problem chosen at random from that exercise.

Solution:

Step 1: Define the probabilities of each student solving a problem.

The probability that student A can solve a problem: P(A) = 60%

P(A) = 60/100 = 0.6

The probability that student B can solve a problem: P(B) = 75%

P(B) = 75/100 = 0.75

Step 2: Find the probability that both A and B can solve a problem (i.e. P(A∩B)).

Since the events are independent, (one student's ability does not affect the other). P(A∩B) can be found by the following formula:

P(A ∩ B) = P(A) × P(B)

= 0.6 × 0.75

P(A ∩ B) = 0.45

Step 3: Use the formula to calculate A or B both occurring.

∴ P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Step 3: Put the values into the above equation.

P(A ∪ B) = 0.6 + 0.75 - 0.45

P(A ∪ B) = 0.90

Thus, the probability that either Student A or Student B can solve a problem chosen at random from the exercise is 0.9 or 90%.