Conditional Probability

Grade 8 · statistics · 95 practice problems · read aloud

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Conditional Probability 🎯

What is Conditional Probability?

Conditional probability is the chance of an event happening, given that another event has already occurred. It's like updating your guess based on new information! We write it as P(A|B), which means "the probability of A given B."

Why it's useful: It helps us make smarter predictions in real life. For example, what's the probability you brought an umbrella, given that you see dark clouds outside?

How to Solve Conditional Probability Problems

  1. Identify the "given": Figure out which event is the condition (this is the event that has already happened).
  2. Find the total for the "given" event: In your sample space, only look at the outcomes where the given event occurred.
  3. Find the favorable outcomes: From your new, smaller group, count the outcomes for the event you're finding the probability for.
  4. Calculate the probability: Divide the favorable outcomes by the total outcomes in your new, smaller group.

Worked Examples

Example 1: The Pizza Party 🍕

A class survey found that 20 students like pepperoni pizza, 15 like cheese, and 5 like both. What is the probability that a student likes cheese, given that they like pepperoni?

Step 1: The "given" is that they like pepperoni. So, we only look at the 20 students who like pepperoni.

Step 2: From this group, the number who also like cheese is 5 (the ones who like both).

Step 3: Probability = (Students who like both) / (Students who like pepperoni) = 5 / 20 = 1/4 or 25%.

Example 2: Deck of Cards 🃏

You draw one card from a standard deck. What is P(Queen | Face Card)?

Step 1: The "given" is that it's a face card (Jack, Queen, King). There are 12 face cards in a deck.

Step 2: From these 12 face cards, how many are Queens? There are 4 Queens (one in each suit).

Step 3: Probability = 4 / 12 = 1/3.

Common Mistakes to Avoid

⚠️ Using the original total: The biggest mistake is using the total number from the original whole group. Remember, the "given" condition shrinks your sample space! Always use the total for the given event as your denominator.

⚠️ Confusing P(A|B) and P(B|A): These are different! P(Rain | Clouds) is not the same as P(Clouds | Rain). Always check which event is the condition.

Tips & Tricks

💡 The "Given" Shrinks the World: Imagine the "given" event creates a new, smaller world. You only care about what happens inside that world now.

💡 Two-Way Tables are Your Friend: Organizing data in a table makes it much easier to see the numbers for the "given" group and the favorable outcomes.

💡 Formula to Know: P(A|B) = P(A and B) / P(B). You'll use this more in higher grades, but it's good to see it now!

How to Practice

  • Create your own surveys with friends (favorite music, sports, etc.) and calculate conditional probabilities from the results.
  • Use a deck of cards or dice. Ask questions like, "What's the probability of rolling an even number, given that the number is greater than 3?"
  • Look for "given" in word problems—it's your keyword that a conditional probability is needed!

Practice problems

6 of the 95, worked through step by step — try them before opening the answer.

1 (3² × 4) ÷ (2³ - 2) = ?

Hint: Remember to follow the order of operations: parentheses first, then exponents, then multiplication and division from left to right.

Show the answer

Answer: 6

  1. Calculate the exponent in the numerator. 3² means 3 × 3 = 9. So numerator becomes: 9 × 4.
  2. Multiply in the numerator. 9 × 4 = 36. So numerator = 36.
  3. Calculate the exponent in the denominator. 2³ means 2 × 2 × 2 = 8. So denominator becomes: 8 - 2.
  4. Subtract in the denominator. 8 - 2 = 6. So denominator = 6.
  5. Divide numerator by denominator. 36 ÷ 6 = 6. Final answer: 6

Let's solve step by step. We have: (3² × 4) ÷ (2³ - 2)

2 P(A) = 2/5, P(B) = 4/9, P(A and B) = 8/45. Are events A and B independent?

Hint: Check if the product of the individual probabilities equals the probability of both events occurring together.

Show the answer

Answer: Yes

  1. Calculate P(A) × P(B) = (2/5) × (4/9) = 8/45.
  2. Compare this to P(A and B) = 8/45.
  3. Since P(A) × P(B) = P(A and B), the events are independent.

The answer is Yes.

3 P(A) = 1/6, P(B) = 1/6, P(A and B) = 1/36. Are events A and B independent?

Hint: Check if the product of the individual probabilities equals the probability of both events occurring together.

Show the answer

Answer: Yes

  1. Calculate P(A) * P(B) = (1/6) * (1/6) = 1/36.
  2. Compare with P(A and B) = 1/36.
  3. Since P(A) * P(B) = P(A and B), the events are independent.

The answer is Yes.

4 P(A) = 2/7, P(B) = 3/7, P(A and B) = 6/49. Are events A and B independent?

Hint: Check if the product of the individual probabilities equals the probability of both events occurring together.

Show the answer

Answer: Yes

  1. Calculate P(A) * P(B) = (2/7) * (3/7) = 6/49.
  2. Compare with P(A and B) = 6/49.
  3. Since 6/49 = 6/49, the condition P(A) * P(B) = P(A and B) holds.
  4. Therefore, events A and B are independent.

The answer is Yes.

5 P(A) = 7/12, P(B) = 9/14, P(A and B) = 3/8. Are events A and B independent?

Hint: Check whether the product of the individual probabilities equals the probability of both events occurring together. If they are not equal, the events are not independent.

Show the answer

Answer: No

  1. Multiply P(A) and P(B): (7/12) × (9/14) = 63/168 = 3/8.
  2. Compare with P(A and B) = 3/8.
  3. Since 3/8 = 3/8, the product equals the joint probability.
  4. Therefore, events A and B are independent.

The answer is Yes.

6 P(A) = 8/15, P(B) = 9/20, P(A and B) = 6/25. Are events A and B independent?

Hint: Check whether the product of the individual probabilities equals the probability of both events occurring together. If they are equal, the events are independent.

Show the answer

Answer: Yes

  1. Multiply P(A) and P(B): (8/15) × (9/20) = 72/300 = 6/25.
  2. Compare with P(A and B) = 6/25.
  3. Since 6/25 = 6/25, the product equals the joint probability.
  4. Therefore, events A and B are independent.

The answer is Yes.

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