# Probability

## Introduction

Probability is used in the real world to determine how likely it is for an event to occur. For example, you can use probability to find out how likely it is for you to flip a heads with a coin or how likely it is for you to roll certain numbers on a die.

Probability of an event = number of ways for it to happen / total number of possible outcomes

Probability is represented as a number between 0 and 1 where 0 means there is no chance of the event happening and 1 means that it is certain this event will happen. If two events have the same probability, they are equally likely.

## Example 1

How likely is it to roll a six sided die and get a 3?

### Solution:

The number of ways for it to happen is 1: you roll a 3.

The total number of possible outcomes is 6: there are six sides on the die.

Probability of getting a 3 = 1/6 = ~ 0.17

## Example 2

The spinner to the left has equally sized colored areas. What is the chance that the spinner lands on green?

### Solution:

The number of ways for it to happen is 3: there are 3 green sections.

The total number of possible outcomes is 8: there are 8 sections in total on the spinner.

Probability of getting green: 3/8 = 0.125

## Example 3

If you flip a coin 50 times, what is the predicted amount of heads you will get?

### Solution:

First, we need to find the probability of getting heads if you flip a coin once.

The number of ways for it to happen is 1: the coin lands on heads.

The total number of possible outcomes is 2: heads and tails.

Probability of getting heads = 1/2 = 0.5

To find the predicted amount of heads, we need to multiply the total number of flips by the probability of getting a head.

50 total flips * 0.5 chance of getting a head = 25 predicted heads

## Example 4

Create a model of a box of marbles where the chance of picking a red marble is 2/5.

### Solution:

Based on the probability given to us, we know that the number of ways to pick a red marble (the numerator) is 2, which means there are 2 red marbles. We also know that the total number of outcomes (the denominator) is 5, meaning there are 5 total possible outcomes or 5 marbles in the bag. Shown to the right is one possible way of answering this question.

## Theoretical vs. Experimental Probabilities

All the probabilities we have calculated so far have been theoretical probabilities, which is what we expect to happen. However, in real life, things donâ€™t always happen that way. For example, if you do a quick experiment and flip a real life coin 2 times, you may get 2 heads and 0 tails instead of the expected 1 heads and 1 tails. Our experimental probability (the actual results of the experiment) did not match our theoretical probability (what we calculated). To get an experimental probability that is closer to the theoretical probability, you must use more trials. If you flip the real life coin 100 times, you will find that the experimental probability will likely be much closer to the theoretical probability.