# Calculating Probability

Posted November 05, 2021

Probability can come in handy when determining the likelihood of a win, projecting sales growth, and using data to predict what will happen in certain circumstances. Knowing the probability of an event can help someone make a wager, decide how much to invest, or make a justified decision one way or another. If you want to determine the chances of something happening then probability is the type of calculation you will need.

This article will explain the definition of probability, the calculation for the probability of single and multiple events, and the difference between odds and probability.

## What is probability?

Probability is a type of mathematics that gives a numerical description of how likely an event is to occur. The numerical description between 0 and 1 that represents the possibility of a certain outcome where 0 indicated the impossibility of an event and 1 indicated the certainty. The probability of a single event is understood as the likelihood of the event occurring divided by the total number of expected outcomes. When multiple events can occur, probability can be determined by breaking down the individual probability of each singular event, then multiplying those probabilities together.

Probability can be applied to a variety of situations, for example meteorologists and weather forecasters use weather patterns to determine the probability of rain, snow, heat etc. Epidemiologists use probability theory to determine the relationship between exposure and health risks. Scientists use probability to determine the risk and reliability of a space shuttle before testing.

Probability can also help marketing companies decide on a marketing strategy for each of their clients based on the data and statistics of each client’s audience behavior. No matter what the application is, probability is a helpful indicator of what is most likely to happen in order to determine what the next steps to take should be.

## Probability

Knowing the probability of an event occurring can help someone make a decision on a wager or decide when to adjust strategies. Since probability is the likelihood of an event occurring based on the ratio of favorable outcomes to the total number of possible outcomes it can be calculated with a simple mathematical formula. Use the following steps to find the probability of a single event.

1. Identify the number of single events or wanted outcomes.
2. Determine the total number of outcomes that are possible.
3. Divide the number of events by the number of possible outcomes.

### 1. Identify the number of single events or wanted outcomes

First you will need to decide what the wanted outcome is that you are trying to calculate. You might want to know the probability of rolling an even number on a die, or of rolling a 6. You might want to know the probability of your favorite team winning the superbowl versus one of two of your favorite teams winning.

Once you know what event you would like to calculate the probability for, figure out how many wanted outcomes are possible. For the dice, you could roll a 2, 4, or 6 to get an even number so there are 3 possible wanted outcomes. Where as if the event you are trying to determine is the probability of rolling a six there is only one wanted outcome.

### 2. Determine the total number of outcomes that can occur

Next, you need to determine the total number of outcomes that can occur including the event you identified from step one. For example if you are rolling a die, there are six total outcomes that can occur because there are six numbers on a die. In both cases—rolling an even number or simple rolling a six—there may be six different outcomes that can occur.

For the football example, as the teams get eliminated in a tournament the probability of your team winning the final game goes up since the total number of possible winners decreases as teams are eliminated. When there are 8 teams left in the tournament the total number of outcomes possible for the chance of becoming the champions would be 8. After that round there would only be 4 teams that could possibly win so the total outcomes possible at that stage in the tournament would be 4.

### 3. Divide the number of events by the number of possible outcomes

To determine the probability of an event, divide the number of favorable events by the total number of possible outcomes. For example, when rolling a die once and landing on a six is considered one event, you would divide the one event by the six possible outcomes that could occur. This gives us the fraction 1/6, so the probability that you will roll a six on the first try is one in six. The probability is a decimal number between zero and one, so for 1 divided by 6 the probability can be written as 0.1667 or as a percent, 16.7% chance.

If there are only 4 teams left in a tournament, a single team has a one in four chance of winning, or 25% chance of winning if you divide 1 by 4. In the next round a single team has a 50% chance of winning since the total number of possible winners is one of the other, meaning 2 possible outcomes total.

## Odds vs. probability

Probability can sound a lot like the concept of odds but in reality they are two different calculations. Odds are the likelihood of an event actually occurring based on the probability of the favorable outcome and the probability of the unfavorable outcome. Odds consider both the probability of an event occurring and the event not occurring, whereas probability only looks at the favorable event and total possible events.

The odds, or chance, of something happening is represented as a percent chance and is calculated by taking the probability of an event occurring and dividing it by the probability of the event not occurring. So in the case of rolling a six on the first try, the probability is 1/6 that you will roll a six, while the probability that you won't roll a six is 5/6. The probability of an event not occurring can also be calculated by subtracting the probability of an event occurring from one, so 1- 1/6 = 5/6.

The odds are calculated by dividing the two probabilities: 1/6 ÷ 5/6 which equals 1/5 (or 20%) chance that you will actually roll a six on the first try. You will need to find the probability of an event before calculating the odds, but once you have the probability the odds can help you determine the actual likelihood of an event occurring.

## Calculate probability with multiple random events

Sometimes you will want to know what the probability of more than one event is. For example you might want to know the probability of rolling doubles, or what the probability of spinning a \$0.99 on the popular game show the “Price is Right” and then consecutively spinning  a \$0.01 to get a dollar sum. Follow the following steps to calculate the probability of multiple events.

1. Determine each event outcome of interest.
2. Calculate the probability of each independent outcome.
3. Multiply all probabilities together.

### 1. Determine each event outcome of interest

First determine the events you would like to calculate the probability of. For example, if you would like to know the probability of rolling a six on two dice simultaneously then the events of interest would be that the first die lands on six and the second die lands on six. The outcomes are both independently random events, and we want to know the probability of those events happening at the same time.

### 2. Calculate the probability of each independent outcome

Next, you will calculate the probability of each independent outcome. Since we are looking at two dice, we will calculate the probability that each die lands on a six separately. Each probability is independent from the other because we are not looking at the simultaneous probability yet, just the probability of a favored outcome on each die separately. For both events there are six sides to each die and only one 6 on each die. The result is that the probabilities are both 1/6, or 0.1667.

### 3. Multiply all probabilities together

To find the probability of simultaneous random events occurring you will need to multiply their probabilities together. In our example of rolling a six on two dice simultaneously we would take the individual probabilities and multiply them together. So the calculation is 1/6 x 1/6 = 1/36 meaning one event out of 36 total outcomes that the dice will land on six at the same time.

It’s important to know that the probability only compares the likelihood of certain events occurring compared to all possible events. Whereas your chances of winning, or odds compare the likelihood of a win compared to the likelihood of a lose.  Let’s say you would then like to calculate the odds of rolling a six on two dice simultaneously.

We would take our formula from odds above, Probability of (Event A)/Probability of (Not Event A) = Odds of Event A.

For rolling double sixes that means we would take the probability of rolling two sixes (1/36) and divide it by the probability of not rolling two sixes (35/36) the result would be 1/35 or 0.0285, meaning that you have a 2.85% chance of rolling two sixes simultaneously on two dice.

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