Introduction

In the realm of probability theory, the ability to calculate the likelihood of different events occurring is crucial. A discrete probability function provides a systematic way to assign probabilities to individual outcomes. In this chapter, we’ll explore how to calculate the probability of specific events using a discrete probability function. By mastering this skill, you’ll gain the foundation to make informed decisions in scenarios where uncertainty plays a role.

Discrete Probability Functions

A discrete probability function, often denoted as $P(X)$, is a mathematical model that assigns probabilities to discrete outcomes of a random variable $X$. Each outcome is associated with a certain probability, and the probabilities for all possible outcomes sum up to 1.

To calculate the probability of a particular event A occurring, we use the formula: $P(A)=\frac{\text{ Number of favorable outcomes for event A }}{\text{ Total number of possible outcomes }}$

Example: Consider a standard six-sided die. The event $\mathrm{A}$ is rolling an even number. There are three favorable outcomes (2, 4 and 6), and the total number of possible outcomes is six. Therefore, the probability of event $\mathrm{A}$ is:

$$P(A)=\frac{3}{6}=\frac{1}{2}$$

Calculating Joint Probabilities

For two events A and B, the joint probability of both events occurring can be calculated using the formula:

$$P(A\cap B)=P(A)\times P(B\mid A)$$

Example: Let’s take the example of drawing two cards from a standard deck of 52 playing cards. Event $A$ is drawing a red card on the first draw, and event $B$ is drawing a red card on the second draw after the first draw resulted in a red card. If we draw without replacement, the probability of event $\mathrm{A}$ is $P(A)=\frac{26}{52}$. After drawing a red card, there are 25 red cards left in a deck of 51 cards, so the probability of event $B$ given that event $A$ has occurred is $P(B\mid A)=\frac{25}{51}$. Therefore, the joint probability of both events is:

$$P(A\cap B)=P(A)\times P(B\mid A)=\frac{26}{52}\times \frac{25}{51}$$