# 2.3: Conditional Probability

What if the probability of a second event is affected by the probability of the first event? This type of probability calculation is known as **conditional probability**.

When working with events that are conditionally probable, you are working with 2 events, where the probability of the second event is conditional on the first event occurring. Say, for example, that you want to know the probability of drawing 2 kings from a deck of cards. As we have previously learned, here is how you would calculate this:

Now let’s assume you are playing a game where you need to draw 2 kings to win. You draw the first card and get a king. What is the probability of getting a king on the second card? The probability of getting a king on the second card can be thought of as a conditional probability. The formula for calculating conditional probability is given as:

Another way to look at the conditional probability formula is as follows. Assuming the first event has occurred, the probability of the second event occurring is:

Let’s work through a few problems using the formula for conditional probability.

*Example 18*

You are playing a game of cards where the winner is determined when a player gets 2 cards of the same suit. You draw a card and get a club . What is the probability that the second card will be a club?

*Solution:*

**Step 1:** List what you know.

First event = drawing the first club

Second event = drawing the second club

**Step 2:** Calculate the probability of choosing a club as the second card when a club is chosen as the first card.

**Step 3:** Write your conclusion.

Therefore, the probability of selecting a club as the second card when a club is chosen as the first card is 24%.

*Example 19*

In the next round of the game, the first person to be dealt a black ace wins the game. You get your first card, and it is a queen. What is the probability of obtaining a black ace?

*Solution:*

**Step 1:** List what you know.

First event = being dealt the queen

Second event = being dealt the black ace

**Step 2:** Calculate the probability of choosing black ace as a second card when a queen is chosen as a first card.

**Step 3:** Write your conclusion.

Therefore, the probability of selecting a black ace as the second card when a queen is chosen as the first card is 7.7%.

*Example 20*

At Bluenose High School, 90% of the students take physics and 35% of the students take both physics and statistics. What is the probability that a student from Bluenose High School who is taking physics is also taking statistics?

*Solution:*

**Step 1:** List what you know.

**Step 2:** Calculate the probability of choosing statistics as a second course when physics is chosen as a first course.

**Step 3:** Write your conclusion.

Therefore, the probability that a student from Bluenose High School who is taking physics is also taking statistics is 39%.

*Example 21*

Sandra went out for her daily run. She goes on a path that has alternate routes to give her a variety of choices to make her run more enjoyable. The path has 3 turns where she can go left or right at each turn. The probability of turning right the first time is . Based on past runs, the probability of turning right the second time is . Draw a tree diagram to represent the path. What is the probability that she will turn left the second time after turning right the first time?

*Solution:*

**Step 1:** List what you know.

**Step 2:** Calculate the probability of choosing left as the second turn when right is chosen as the first turn.

**Step 3:** Write your conclusion.

Therefore, the probability of choosing left as the second turn when right was chosen as the first turn is 33%.

**Points to Consider**

- How does a permutation differ from a combination?
- How are tree diagrams helpful for determining probabilities?

**Vocabulary**

- Combinations
- The number of possible arrangements of objects without regard to order and without repetition selected from a distinct number of objects .

- Conditional probability
- The probability of a particular dependent event, given the outcome of the event on which it depends.

- Factorial function (!)
- To multiply a series of consecutive descending natural numbers.

- Fundamental Counting Principle
- If an event can be chosen in different ways and another independent event can be chosen in different ways, the probability of the 2 events occurring is .

- Permutations
- The number of possible arrangements in an ordered set of objects, where the number of objects and the number of objects selected.

- Tree diagrams
- A way to show the outcomes of simple probability events, where each outcome is represented as a branch on a tree.