# Introduction to Probability

Probability is a critical component of statistical analysis and serves as a way to determine the likelihood of an event occurring. This post will provide a brief introduction into some of the principles of probability.

Probability

There are several basic probability terms we need to cover

• events
• trial
• mutually exclusive and exhaustive

Events are possible outcomes. For example, if you flip a coin, the event can be heads or tails. A trial is a single opportunity for an event to occur. For example, if you flip a coin one time this means that there was one trial or one opportunity for the event of heads or tails to occur.

To calculate the probability of an event you need to take the number of trials an event occurred divided by the total number of trials. The capital letter “P” followed by the number in parentheses is always how probability is expressed. Below is the actual equation for this

Number of trial the event occurredTotal number of trials = P(event)

To provide an example, if we flip a coin ten times and we recored five heads and five tails, if we want to know the probability of heads this is the answer below

Another term to understand is mutually exclusive and exhaustive. This means that events cannot occur at the same time. For example, if we flip a coin, the result can only be heads or tails. We cannot flip a coin and have both heads and tails happen simultaneously.

Joint Probability

There are times were events are not mutually exclusive. For example, lets say we have the possible events

1. Musicians
2. Female
3.  Female musicians

There are many different events that came happen simultaneously

• Someone is a musician and not female
• Someone who is female and not a musician
• Someone who is a female musician

There are also other things we need to keep in mind

• Everyone is not female
• Everyone is not a musician
• There are many people who are not female and are not musicians

We can now work through a sample problem as shown below.

25% of the population are musicians and 60% of the population is female. What is the probability that someone is a female musician

To solve this problem we need to find the joint probability which is the probability of two independent events happening at the same time. Independent events or events that do not influence each other. For example, being female has no influence on becoming a musician and vice versa. For our female musician example, we run the follow calculation.

P(Being Musician) * P(Being Female) = 0.25 * 0.60 = 0.25 = 15%

From the calculation, we can see that there is a 15% chance that someone will be female and a musician.

Conclusion

Probability is the foundation of statistical inference. We will see in a future post that not all events are independent. When they are not the use of conditional probability and Bayes theorem is appropriate.