# Probability

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## Biostatistics and epidemiology

#### Biostatistics

Introduction to biostatistics
Statistical probability distributions
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Outcomes of statistical hypothesis testing

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### Probability

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The probability of having either event X or event Y can be calculated by (adding up/subtracting) the two probabilities, assuming that X and Y are mutually exclusive.

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#### Content Reviewers:

Rishi Desai, MD, MPH

#### Contributors:

Probability is the chance that an event or outcome will occur, and it’s calculated by dividing the number of times an event happened by the number of times the event could have happened.

For example, let’s say you have one six-sided die and you want to know the probability of rolling a certain number, like a three.

Typically, probability is written with a capital P, and P of A represents the probability of “event A” happening.

In this situation, event A is rolling a 6. Since a die has six sides, there are six possible numbers you could roll, so the probability of rolling a three is 1 divided by 6, or 0.167.

Probability can be written as a decimal or as a percent, so the chance of rolling a three is 0.167 times 100 or 16.7%.

Now, there are eight basic rules in probability.

The first rule states that the probability of event A can range anywhere from 0 - or 0% - to 1 - or 100%.

The larger the probability is, the higher the chance that the event will occur.

The second rule states that the sum of the probabilities of all possible outcomes has to equal 1.

For example, the probability of rolling each side of the die is 0.167, and when we add up 0.167 six times, it equals 1.

Sometimes we might want to find the probability that an event won’t occur - like if we wanted to figure out the probability of not rolling a three.

The probability of an event not occurring is called the complement, and it’s written as the probability of the event, except it has a prime symbol - which is just an apostrophe.

The third rule of probability states that the probability that an event doesn’t occur is 1 minus the probability that it does occur.

So, the probability of not rolling a three is 1 minus 0.167, or 0.833.

Turning that around, it also means that the probability of the event occurring equals 1 minus the complement. This is helpful in situations where we want to figure out the probability that an event occurs, but only know the probability that the event won’t occur.

Rule 4 has to do with finding the probability of two or more events happening, which is called a compound event.

For example, let’s say we want to know the probability of rolling a 3 or rolling a 5. In this case, rolling a three is the first event, or event A, and rolling a 5 is the second event, or event B, so the probability of A or B is a compound event.

Now, there are two types of compound events, and the first type is the union of two events, which is the probability of A or B occurring.

Typically it’s written with the union symbol, which looks like a small U.

In a union of two events, the two events can either be disjoint - or mutually exclusive - or not disjoint - or not mutually exclusive.

So, if event A is rolling a 3 and event B is rolling a 5, then A and B are disjoint events, because it’s impossible to roll a 3 and a 5 number at the same time.

Disjoint events are often represented by two circles - one for event A and one for event B - sitting side by side, with no overlapping area.

Rule 4 is also called the addition rule and it states that the probability of two disjoint events is the sum of the first event plus the second event.

So, the probability of rolling a 3 is 0.167 and the probability of rolling a 5 is 0.167, so the probability of either rolling a 3 or a 5 is 0.167 plus 0.167, or 0.334, or 33.4%.

If two events are not disjoint, then the two events can occur at the same time.

For example, if event A is rolling a number less than or equal to 2 - so rolling a 1 or 2 - and event B is rolling an even number, then A and B are not disjoint events, because it’s possible to roll a 2, which is an even number that’s less than or equal to 2.

Not disjoint events are often represented by two overlapping circles, and the overlapping area is the probability of both events occurring at the same time.

In this example, the overlapping area is the probability of rolling a 2; the non-overlapping area in circle A is the probability of rolling a 1 - which is a number that is less than or equal to 2, but not an even number; and the non-overlapping area in circle B is the probability of rolling a 4 or 6, which are even numbers but not numbers that are less than or equal to 2.

It’s a bit more complicated to calculate the probability for not disjoint events because you have to take into account the overlapping area. So let’s break it down.

The probability of rolling a number less than or equal to 2 is 2 over 6, or 0.33, and the probability of rolling an even number - so a 2, 4, or 6 - is 3 over 6, or 0.5. If we use the basic addition rule, the probability of events A or B is 0.33 plus 0.5, which is 0.83 - or 83%.

But this number is actually higher than the true probability for events A or B, because the overlapping area is counted twice.

To fix this, we need to subtract the probability of the overlapping area from the sum of the two individual probabilities.

So Rule 5 states that the probability for two not disjoint events equals the sum of the probability of event A and the probability of event B, minus the probability of event A and B together.

For example, let’s say the probability of rolling an even number that’s less than or equal to 2 is 0.167. If the probability of event A is 0.33 and the probability of event B is 0.5, then the probability of event A or event B occurring is 0.33 plus 0.5 minus 0.167, which equals 0.663, or 66.3%.

One important thing to notice is that the addition rule for disjoint events is actually the same as the addition rule for not disjoint events - except with disjoint events there is no overlapping area so you just subtract 0, and this part isn’t usually included in the equation.

Now let’s switch gears and talk about the situation when you want to figure out the probability that both event A and event B will occur - or in other words, the probability of the overlapping area. This is the intersection of two events, and it’s the second type of compound event.

Typically, the intersection of two events is written with an intersection symbol, which looks like an upside down U.