Probability Calculator

Probability of Two Events

Probability is simply the science of measuring how likely something is to happen. It helps answer questions like:

• What are the chances of winning a game?
• How likely is it to rain tomorrow?
• What are the odds of drawing a specific card?
• What is the chance two things happen together?

Probability is always expressed as a number between 0 and 1:

• 0 = Impossible (it will never happen)
• 1 = Certain (it will always happen)
• Values between 0 and 1 represent different levels of likelihood.

For example:

• A probability of 0.75 means there is a 75% chance something happens.
• A probability of 0.10 means there is only a 10% chance.

The basic probability formula is:$$Probability = \frac{\text{Desired Outcomes}}{\text{Total Possible Outcomes}}$$Probability=Total Possible OutcomesDesired Outcomes​

However, probability becomes more interesting when we look at multiple events. The relationship between events determines whether we should add, subtract, or multiply probabilities.

Events can be:

• Independent — one event does not affect the other
• Dependent — one event changes the outcome of another
• Mutually exclusive — two events cannot happen at the same time

This calculator helps you explore different probability scenarios, including:

• The chance an event does not happen
• The probability that two events happen together
• The probability that at least one event happens
• The probability that exactly one event happens

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Complement of an Event

The complement of an event is the probability that something does not happen.

If event A has a probability of:$$P(A)$$P(A)

then the chance that event A does not occur is:$$P(A’) = 1 – P(A)$$P(A′)=1−P(A)

Example

Imagine Bob has a:$$65\%$$65%

chance of forgetting his homework.

The probability that Bob does complete his homework is:$$P(A’) = 1 – 0.65$$P(A′)=1−0.65 $$P(A’) = 0.35$$P(A′)=0.35

So Bob has a:$$35\%$$35%

chance of turning in his homework.

The same rule applies to any event. If you know the probability something happens, subtracting it from 1 gives you the probability that it does not happen.

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Probability of A and B (Intersection)

The intersection of two events means both events happen.

It is written as:$$P(A \cap B)$$P(A∩B)

or:$$P(A \text{ AND } B)$$P(A AND B)

Think of it as the overlap between two possibilities.

For example:

• Event A = Rolling a 6 on one dice roll
• Event B = Rolling another 6 on a second dice roll

Since both rolls are independent:$$P(A \cap B)=P(A)\times P(B)$$P(A∩B)=P(A)×P(B)

For a six-sided die:$$\frac{1}{6}\times\frac{1}{6}$$61​×61​ $$=0.0278$$=0.0278

There is about a:$$2.78\%$$2.78%

chance of rolling two sixes in a row.

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Dependent Events

Not all events are independent.

Sometimes one event changes the probability of another event.

For example, imagine a bag containing:

• 7 black marbles
• 3 blue marbles

There are 10 total marbles.

The chance of picking a blue marble first is:$$P(A)=\frac{3}{10}$$P(A)=103​

After removing that blue marble, only 9 marbles remain.

Now the probability of picking a black marble becomes:$$P(B|A)=\frac{7}{9}$$P(B∣A)=97​

This is called conditional probability — the probability of an event happening after another event has already occurred.

The probability of picking blue and then black is:$$P(A \cap B)=P(A)\times P(B|A)$$P(A∩B)=P(A)×P(B∣A) $$=\frac{3}{10}\times\frac{7}{9}$$=103​×97​ $$=0.2333$$=0.2333

There is a:$$23.33\%$$23.33%

chance of drawing a blue marble followed by a black marble.

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Probability of A or B (Union)

The union of two events means one or both events happen.

It is written as:$$P(A \cup B)$$P(A∪B)

or:$$P(A \text{ OR } B)$$P(A OR B)

There are two situations.

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When Events Cannot Both Happen

Some events are mutually exclusive.

Example:

Rolling a single die:

• Event A = Rolling an even number
• Event B = Rolling an odd number

A number cannot be both even and odd.

The formula is:$$P(A \cup B)=P(A)+P(B)$$P(A∪B)=P(A)+P(B)

For a die:$$\frac{3}{6}+\frac{3}{6}=1$$63​+63​=1

The probability is:$$100\%$$100%

because every roll is either even or odd.

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When Events Can Overlap

Sometimes events can happen at the same time.

The formula becomes:$$P(A \cup B)=P(A)+P(B)-P(A\cap B)$$P(A∪B)=P(A)+P(B)−P(A∩B)

The overlap must be removed because it was counted twice.

Example

Rolling a die:

Event A:

• Rolling an even number

$$P(A)=\frac{3}{6}$$P(A)=63​

Event B:

• Rolling a multiple of 3

$$P(B)=\frac{2}{6}$$P(B)=62​

The overlap:$$P(A\cap B)=\frac{1}{6}$$P(A∩B)=61​

(rolling a 6)

Therefore:$$P(A\cup B)=\frac{3}{6}+\frac{2}{6}-\frac{1}{6}$$P(A∪B)=63​+62​−61​ $$=\frac{4}{6}$$=64​ $$=66.67\%$$=66.67%

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Exclusive OR (XOR)

Sometimes we want to know the probability that one event happens, but not both.

This is called an exclusive OR.

The formula is:$$P(A XOR B)=P(A)+P(B)-2P(A\cap B)$$P(AXORB)=P(A)+P(B)−2P(A∩B)

Example:

Imagine two candy bowls:

• Bowl A contains Reese’s
• Bowl B contains Snickers

A child can choose:

• Reese’s only
• Snickers only

but not both.

The formula removes the chance of choosing both candies.

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Normal Distribution

The normal distribution, also called the bell curve, is one of the most common probability patterns in statistics.

Many real-world measurements follow this pattern:

• Heights
• Test scores
• Product measurements
• Temperatures
• Manufacturing data

Most values cluster around the average, while fewer values appear farther away.

A normal distribution is described using:

• Mean (μ) — the average value
• Standard deviation (σ) — how spread out the data is

When:$$\mu=0$$μ=0

and:$$\sigma=1$$σ=1

the distribution is called a standard normal distribution.

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Using Z-Scores

To find probabilities in a normal distribution, values are converted into Z-scores.

The formula is:$$Z=\frac{x-\mu}{\sigma}$$Z=σx−μ​

where:

• x = value being measured
• μ = average
• σ = standard deviation

Example

Suppose college students have an average height of:$$68 \text{ inches}$$68 inches

with a standard deviation of:$$4 \text{ inches}$$4 inches

A height of 60 inches becomes:$$Z=\frac{60-68}{4}$$Z=460−68​ $$Z=-2$$Z=−2

A height of 72 inches becomes:$$Z=\frac{72-68}{4}$$Z=472−68​ $$Z=1$$Z=1

Using the normal distribution table, we can determine the probability of a student being between those heights.

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Confidence Intervals and Probability

Probability is also used in statistics to estimate unknown values.

A confidence interval provides a range where a population value is likely to fall.

For example:

“We are 95% confident the average customer rating is between 4.5 and 4.8 stars.”

Instead of giving a single estimate, confidence intervals show the possible range of realistic values.

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This calculator helps make probability easier by turning formulas into practical answers — whether you’re calculating odds, analyzing data, or simply wondering “what are the chances?”.