Chapters

## Probability

**biology, finance, economics**and more. In order to understand what probability is, we first need to go through two very important concepts: random variable and sample space.

A random variable is a variable whose outcome is **unknown.** A die toss is a classic example of a random variable: the outcome is unknown until the moment the die is tossed.

All possible outcomes of the random variable makeup it’s **sample space,** whose notation you can see in the image above. An event, on the other hand, is simply one or more outcomes in the sample space.

**Probability,** then, can be defined as the **likelihood** of an event occurring. The simplest formula for probability is located below.

## Probability Distribution

A probability distribution is a **visualization** of all possible outcomes of a random variable along with the probability of each value occurring. Let’s revisit the die roll example: say you want to know the probability of having a sum of 2 when rolling **2** **dice.**

1 | 2 | 3 | 4 | 5 | 6 | |

1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |

2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |

3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |

4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |

5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |

6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |

Next, we figure out the probability of each sum of dice. Notice that the **minimum sum** we have is 2, because 1+1=2, and the maximum sum is 12. Each diagonal has the same sum. For example, take a look at the first two diagonals.

This makes it easy for us to come up with the probabilities of each sum occurring, which is the **probability distribution.**

Sum | Probability (Fraction) | Probability (Decimal) |

2 | 1/36 | 0.028 |

3 | 2/36 | 0.056 |

4 | 3/36 | 0.083 |

5 | 4/36 | 0.111 |

6 | 5/36 | 0.139 |

7 | 6/36 | 0.167 |

8 | 5/36 | 0.139 |

9 | 4/36 | 0.111 |

10 | 3/36 | 0.083 |

11 | 2/36 | 0.056 |

12 | 1/36 | 0.028 |

This gives us the following probability distribution.

## Standard Normal Distribution

We looked at a simple probability distribution for rolling 2 dice - however, what if we wanted to know the probability distribution of another random variable. For example, the distribution for a coin toss or the distribution for the amount of cars on the highway? Every random variable has its own probability distribution. While there are **many different** kinds, we can focus on a standard normal distribution.

A **normally** distributed variable is one whose probability distribution is shaped like a bell, which is why they are also known as “bell-shaped” distributions. Dice rolls, when rolled enough times, actually have normal distributions.

The only difference between a normal and standard normal distribution is **the scale** of the numbers. While a normal distribution has the values of the x-axis as raw scores, the standard normal distribution has it’s values in z-scores.

## Binomial Distribution

Binomial distributions are those whose random variables have only **two** **outcomes.** Some examples are listed in the table below.

Example | Outcomes |

Result of an interview | Success or Failure |

Coin toss | Heads or Tails |

Exam score | Pass or Fail |

The parameters for a **binomial** distribution are below.

The elements of the probability **formula** are explained below.

Element | Description |

n | Number of trials (called Bernoulli trials) |

k | Number of successes in n trails |

p | Probability of success |

q | Probability of failure (q = 1-p) |

## Poisson Distribution

A poisson distribution, on the other hand, is a distribution that deals with **time.** Some examples of random variables with poisson distributions are given below.

Example | Example Outcome |

Number of calls at a call centre per hour | 15 calls per hour |

Number of hours a lightbulb will function | 100 hours |

Number of clients at a store during the day | 200 clients per day |

To calculate the probability of a poisson distribution, we use the **characteristics** of the parameters, listed below.

Element | Description |

Mean rate (mean # occurrences per interval) | |

e | Natural |

k | Probability of success |

## Problem 1

A call centre for an amusement park has been experiencing many **complaints** from customers who are unhappy about long waiting times. The call centre has a policy that states that if you wait for more than half an hour, you will get a gift card to the amusement park.

Wanting to improve call times, the call centre wants to know how likely it is that **32 customers** will call in **4 hours** given that the average number of calls per hour is 5. First, state which distribution you would use.

## Solution 1

In order to solve this, first you should understand what type of distribution we’re dealing with here. The distribution we would use to solve this is a **Poisson** distribution. The reason for this is that we’re dealing with intervals of time.

Next, we need to outline what each **parameter** is, shown in the picture below.

## Problem 2

Next, continuing from problem 1, you want to calculate the probability of 32 people calling in four hours given that the **average** amount of calls per hour is 5. Solve this scenario.

## Solution 2

While it may seem there is more than one way to solve this, there is only one way. Because we want to compare the amount of people calling in 4 hours and not per hour, we can simply multiply **5 by 4**. This is because each **interval** of 1 hour has an average of 5 calls.

We **cannot** solve this problem by **dividing** the number of people in 4 hours by 4 hours. In other words, find how many people are calling per hour if the total is 32 in 4 hours. This gives us a different probability.

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