Conditional Probability

Grade 10 · mathematics · 81 practice problems · read aloud

🔊 Listen to this explanation

🎯 What is Conditional Probability?

Conditional probability is the probability of an event happening given that another event has already occurred. We write it as P(A|B), which means "the probability of A given B."

This is super useful in real life! For example: What's the probability you'll bring an umbrella given that it's cloudy outside? The condition (clouds) changes the probability.

📝 Step-by-Step Guide

  1. Identify the two events: Event A (what you want) and Event B (the condition)
  2. Use the formula: P(A|B) = P(A and B) ÷ P(B)
  3. Calculate both probabilities from the problem
  4. Divide and simplify your answer

🔍 Visual Examples

Example 1: The Dice Roll

Problem: You roll a fair 6-sided die. What's the probability you roll a 4 given that you rolled an even number?

Step 1: A = rolling 4, B = rolling even number

Step 2: P(A and B) = P(4) = 1/6

Step 3: P(B) = P(even) = 3/6 = 1/2

Step 4: P(4|even) = (1/6) ÷ (1/2) = (1/6) × 2 = 1/3

Example 2: Classroom Survey

Problem: In a class of 30 students: 18 play sports, 12 play music, and 6 do both. What's the probability a student plays sports given that they play music?

Step 1: A = plays sports, B = plays music

Step 2: P(A and B) = 6/30 = 1/5

Step 3: P(B) = 12/30 = 2/5

Step 4: P(sports|music) = (1/5) ÷ (2/5) = 1/2

⚠️ Common Mistakes

Swapping A and B: P(A|B) is NOT the same as P(B|A)! "Given that" tells you which event is the condition.

Forgetting the condition: Don't use the total sample space - use only the "given" group for your denominator.

Formula confusion: Remember it's P(A and B) ÷ P(B), not P(A) ÷ P(B).

💡 Tips & Tricks

Memory aid: "The given goes on the bottom!" The condition (B) is always in the denominator.

Venn diagrams: Draw them! P(A|B) is the overlap ÷ the B circle.

Tree diagrams: Great for multi-step conditional probability problems.

Real-world thinking: Ask "If I know B happened, how does that change A's chances?"

🎯 Practice Suggestions

  • Start with simple dice/card problems to master the formula
  • Create 2-way tables from survey data and find conditional probabilities
  • Practice identifying which event is the "given" in word problems
  • Try real-life scenarios: "If it's raining, what's the probability the game is cancelled?"
  • Mix with independent events to see the difference

Practice problems

6 of the 81, worked through step by step — try them before opening the answer.

1 log₂(16) = ?

Hint: Think about what power you need to raise the base to in order to get the number inside the logarithm.

Show the answer

Answer: 4

  1. Understand what the logarithm means. log base 2 of 16 means: "2 raised to what power equals 16?" So we write: 2^x = 16.
  2. Express 16 as a power of 2. 16 = 2 * 2 * 2 * 2 = 2^4.
  3. Substitute 16 with 2^4 in the equation. 2^x = 2^4.
  4. Since the bases are the same (base 2), the exponents must be equal. Therefore, x = 4.
  5. Conclusion. log base 2 of 16 = 4. Final answer: 4

We are solving: log base 2 of 16.

2 P(A) = 0.6, P(B) = 0.5, P(A∩B) = 0.3. Find P(A|B).

Hint: Conditional probability P(A|B) focuses only on the part of event A that lies within event B. Use the formula that relates the intersection to the probability of the given condition.

Show the answer

Answer: 0.6

  1. Write the conditional probability formula: P(A|B) = P(A∩B) / P(B)
  2. Substitute the given values: P(A|B) = 0.3 / 0.5
  3. Perform the division: 0.3 ÷ 0.5 = 0.6
  4. The conditional probability P(A|B) is 0.6.

3 P(A) = 0.6, P(B) = 0.6, P(A∩B) = 0.36. Find P(A|B).

Hint: Conditional probability P(A|B) is found by dividing the probability that both events occur by the probability of the event that is given. Focus on the formula that relates these two probabilities.

Show the answer

Answer: 0.6

  1. Write the conditional probability formula: P(A|B) = P(A∩B) / P(B)
  2. Substitute the given values: P(A|B) = 0.36 / 0.6
  3. Perform the division: 0.36 ÷ 0.6 = 0.6
  4. The conditional probability P(A|B) is 0.6.

4 P(A) = 0.72, P(B) = 0.45, P(A∩B) = 0.27. Find P(A|B).

Hint: Conditional probability P(A|B) focuses only on the portion of event A that lies within event B. Use the formula that relates the intersection probability to the probability of the given condition.

Show the answer

Answer: 0.6

  1. Write the conditional probability formula: P(A|B) = P(A∩B) / P(B)
  2. Substitute the given values: P(A|B) = 0.27 / 0.45
  3. Perform the division: 0.27 ÷ 0.45 = 0.6
  4. The conditional probability P(A|B) is 0.6.

5 P(A) = 0.36, P(B) = 0.6, P(A∩B) = 0.216. Find P(A|B).

Hint: Conditional probability P(A|B) asks: given that B has occurred, what fraction of B's probability is shared with A? Use the formula that relates the intersection to the probability of the condition.

Show the answer

Answer: 0.36

  1. Write the conditional probability formula: P(A|B) = P(A∩B) / P(B)
  2. Substitute the given values: P(A|B) = 0.216 / 0.6
  3. Perform the division: 0.216 ÷ 0.6 = 0.36
  4. The conditional probability P(A|B) is 0.36.

6 P(A) = 0.88, P(B) = 0.75, P(A∩B) = 0.66. Find P(A|B).

Hint: Conditional probability P(A|B) asks: given that B has occurred, what fraction of B's probability is shared with A? Use the formula that divides the intersection probability by the probability of the given event.

Show the answer

Answer: 0.88

  1. Write the conditional probability formula: P(A|B) = P(A∩B) / P(B)
  2. Substitute the given values: P(A|B) = 0.66 / 0.75
  3. Perform the division: 0.66 ÷ 0.75 = 0.88
  4. The conditional probability P(A|B) is 0.88.
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