Learn How to Calculate Probability

Probability is used in many areas of life, including weather forecasting, investing, sports betting and more! Probability is defined as the likelihood of an event (or more than one event) occurring. Probability represents the possibility of obtaining a specific outcome. In today’s post, we will look at how to calculate probability, first looking at the probability formula, then learning how to adjust the formula for single events, independent events, mutually exclusive events and finally, conditional probability.

The probability formula

When we calculate probability, we use a formula which defines the likelihood of the event happening.

The probability formula:

Number of desired outcomes

Probability = ______________________

Total number of outcomes

Another way to write out the formula is this:

Probability = Number of desired outcomes / Total number of outcomes

Or P(A) = f / N

P(A) = Probability of an event (Event A) occurring

f = The number of ways an event can occur (frequency)

N = Total number of possible outcomes

Let’s apply this formula to a scenario

We’ll analyse the probability of landing on an even number when rolling a die.

The desired outcome is landing on an even number and there are 3 even numbers on a die.

The total number of possible outcomes is 6 as there are 6 numbers on a die.

In this instance:

Probability = Number of desired outcomes/Total number of outcomes = 3/6

The possibilities of answers range from 0 to 1. If something has a probability of 0, then it is impossible and if something has a probability of 1, then it is certain.

Calculating the probability of a single event

These are the steps to determining single-event probability:

Determine a single event with a single outcome - The very first step in solving a probability problem is determining the probability you want to work out. It can be an event (for example, the probability of rolling a specific number on a die). The event will need to have at least one possible outcome.
Work out the total number of possible outcomes - Next, you will determine the number of outcomes that could occur from the event you chose in step one. Using the example of rolling a die, there are six total outcomes that could occur (there are six numbers on a die). In the single event of rolling a die and landing on the number 3, there may be six different outcomes that could occur.
Divide the number of desired events by the number of possible outcomes - Once you have determined the single event and its corresponding outcomes, divide the total number of desired events by the total number of possible outcomes. In our example of rolling a die and landing on 3, this is considered one event. You would then divide this one event by the six possible outcomes that could occur.

The calculation is as follows:

Probability= Number of desired outcomes/ Total number of outcomes = 1/6

So, the probability that you will roll a 3 is one in six.

Calculating the probability of independent events

Independent events are events that are not affected or dependent on other events. If we roll two dice, the outcome of each roll has no effect on the other, because they are independent events. The steps to finding the probability of multiple or independent events are similar to the steps used to calculate a single event.

There are a few additional steps to reach the final solution, so the formula for calculating the probability of two events looks like this:

P(A and B) = P(A) x P(B)

Where:

P(A and B) = The probability of both events A and B occurring

P(A) = The probability of event A

P(B) = The probability of event B

These are the steps to determine the probability of multiple events:

Determine each event you want to calculate - The first step is to determine each of the events you want to calculate. For example, let’s calculate the probabilities of rolling a 6 on two dice. Rolling each die separately constitutes one event. We will calculate the probabilities of these two events occurring simultaneously.
Work out the probability of each event - Next, we will calculate the probability of rolling the number 6 on a die and then the probability of rolling a 6 on the other die. The probability for each event is one in six (using the calculations we demonstrated in determining the probability of a single event). Using these results, we will find out the probability of these two independent events occurring at the same time.
Multiply all probabilities together - The final step is to multiply each probability together to determine the total probability for all events that can occur. Using our example of rolling the dice, we will calculate the total probability by multiplying the one in six chances we calculated. Take a look at the calculation below to see how this comes together:

P(A and B) = ⅙ x ⅙ = 1/36

Therefore, there is a one in 36 chance of rolling 6 on one die at the same time as rolling 6 with the other die.

Calculating the probability of mutually exclusive events

Mutually exclusive events are two or more events that cannot take place at the same time. Let’s look at the example of rolling a die and landing on an even or odd number. You cannot land on both at the same time, so these two events are mutually exclusive.

The formula for calculating the probability of mutually exclusive events looks like this:

P(A or B) = P(A) + P(B)

The probability of landing on an even number is 3 in 6.

The probability of landing on an odd number is also 3 in 6.

Therefore, the probability of landing on an even or odd number is:

3/6 + 3/6 = 6/6 = 1

Since landing on an even number or an odd number covers all the possible outcomes, the probabilities add up to 1.

Calculating conditional probability

Conditional probability is the probability of an event occurring based on the results of another event. Let’s look at the example of picking sweets out of a bag. The bag only contains 3 strawberry sweets and 4 lemon sweets. Once we have selected the first sweet, the probabilities change for the second pick, based on the outcome of the first pick.

The probabilities of picking each type of sweet are:

P(Strawberry) = 3/7

P(Lemon) = 4/7

If we picked out one strawberry sweet in our first pick, there would be 2 strawberry sweets and 4 lemon sweets left.

The probabilities would now be:

P(Strawberry) = 2/6

P(Lemon) = 4/6

If instead we picked out one lemon sweet, there would be 3 strawberry sweets and 3 lemon sweets left.

The probabilities would now be:

P(Strawberry) = 3/6

P(Lemon) = 3/6

These probabilities were calculated based on what has already occurred.