Introduction to Conditional Probability

What is probability?

Probability is a mathematical concept that quantifies the likelihood or chance of an event occurring. It is a measure of the degree of certainty we have about the occurrence of an event, based on available information, past experiences, or assumptions. Probability is expressed as a number between 0 and 1, inclusive. A probability of 0 indicates that an event is impossible, while a probability of 1 signifies that an event is certain to occur. The closer the probability is to 1, the more likely the event is to happen, and the closer it is to 0, the less likely it is.

To calculate the probability of a specific outcome, we can use the following formula:

Let's consider an example with two dice. Each die has six faces, numbered 1 through 6. When we roll two dice, there are a total of $6 \times 6 = 36$ possible outcomes, since each die can land on any of its six faces independently.

Now, let's say we want to find the probability of rolling a sum of 7 with the two dice. To do this, we need to identify the favorable outcomes (the combinations of rolls that result in a sum of 7) and divide that number by the total number of possible outcomes (36).

The favorable outcomes for rolling a sum of 7 are:

• Die 1 rolls a 1, and Die 2 rolls a 6.

• Die 1 rolls a 2, and Die 2 rolls a 5.

• Die 1 rolls a 3, and Die 2 rolls a 4.

• Die 1 rolls a 4, and Die 2 rolls a 3.

• Die 1 rolls a 5, and Die 2 rolls a 2.

• Die 1 rolls a 6, and Die 2 rolls a 1.

In the following table, the first row and the first column (in gray) show the result of each of the dice. The colored numbers show the sum of both numbers.

There are six favorable results (in yellow) out of a total of 36 possible results (all colored boxes).

So, the probability of rolling a sum of 7 with two dice is:

What is conditional probability?

Conditional probability is a measure of the likelihood of an event occurring, given that another event has already occurred. It helps us understand the relationship between two events and how the occurrence of one event affects the probability of the other. Conditional probability is denoted as P(A|B), which reads as "the probability of event A happening, given that event B has occurred."

To calculate the conditional probability of event A happening, given that event B has occurred, we use the following formula:

Where:

• $P(A|B)$ represents the conditional probability of event A occurring, given that event B has occurred.

• $P(A \cap B)$ represents the probability of both events A and B occurring.

• $P(B)$ represents the probability of event B occurring.

To better understand this concept, let's look at some examples:

Playing cards example

Let's consider an example to understand the concept of conditional probability. Suppose we have a deck of 52 playing cards. The deck contains 26 red cards (13 hearts and 13 diamonds) and 26 black cards (13 clubs and 13 spades).

We are interested in finding the conditional probability of drawing a heart, given that we have already drawn a red card.

• Event A: Drawing a heart (13 hearts in the deck)

• Event B: Drawing a red card (26 red cards in the deck)

First, let's find $P(B)$ the probability of drawing a red card. There are 26 red cards in a deck of 52 playing cards:

Next, let's find $P(A \cap B)$, the probability of drawing a heart and drawing a red card. Since all hearts are red cards, this is the same as the probability of drawing a heart. There are 13 hearts in the deck:

Now, we can calculate the conditional probability of drawing a heart, given that we have already drawn a red card:

So, the conditional probability of drawing a heart, given that we have already drawn a red card, is 1/2 or 50%.

The school example

Let's consider another example to illustrate the concept of conditional probability. Suppose a school has a total of 500 students, out of which 300 students play sports and 200 students are in the school choir. There are 150 students who participate in both sports and choir. We are interested in finding the conditional probability of a student being in the choir, given that they play sports.

• Event A: Student is in the choir

• Event B: Student plays sports

First, let's find $P(B)$, the probability of a student playing sports. There are 300 students who play sports out of a total of 500 students:

Next, let's find $P(A \cap B)$, the probability of a student being in the choir and playing sports. There are 150 students who participate in both sports and choir:

Now, we can find the conditional probability of a student being in the choir, given that they play sports:

So, the conditional probability of a student being in the choir, given that they play sports, is 0.5 or 50%. This means that if we already know a student plays sports, there is a 50% chance that they are also in the choir.

Congratulations, you have gained expertise in understanding conditional probabilities!