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Conditional Probability

Grade 10 · Mathematics · Worksheet 1

  1. P(A|B) = P(A∩B) / P(B) where P(A) = 0.3, P(B) = 0.6, P(A∩B) = 0.18 = ? Answer: ______________
  2. log₂(16) = ? Answer: ______________
  3. P(A|B) = P(A∩B) / P(B) where P(A) = 0.7, P(B) = 0.8, P(A∩B) = 0.56 = ? Answer: ______________
  4. Mere runs a diagnostic clinic that tests for a rare genetic condition. In the general population, 2% of people have the condition. The test has a 96% true positive rate (correctly identifies those with the condition) and a 4% false positive rate (incorrectly flags those without the condition). If a randomly selected person from the population tests positive for the condition, what is the probability that they actually have the condition? Express your answer as a percentage rounded to two decimal places. Answer: ______________
  5. P(A) = 0.81, P(B) = 0.36, P(A∩B) = 0.2916. Find P(A|B). Answer: ______________
  6. Matiu runs a community garden. He surveyed 240 members to find out their gardening preferences. He found that 144 members grow vegetables, 108 members grow flowers, and 72 members grow both vegetables and flowers. If a randomly selected member from the survey grows vegetables, what is the probability that they also grow flowers? Express your answer as a simplified fraction. Answer: ______________
  7. Isabella volunteers at a local animal shelter. She recorded data on 200 adopted pets. Of these, 120 were dogs and 80 were cats. Among the dogs, 72 were adopted by families with children. Among the cats, 27 were adopted by families with children. If a randomly selected adopted pet was adopted by a family with children, what is the probability that it was a dog? Express your answer as a simplified fraction. Answer: ______________
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Answer Key & Explanations

Conditional Probability · Grade 10 · Worksheet 1

  1. P(A|B) = P(A∩B) / P(B) where P(A) = 0.3, P(B) = 0.6, P(A∩B) = 0.18 = ? Answer: 0.3 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.18 / 0.6 Perform the division: 0.18 ÷ 0.6 = 0.3 The conditional probability P(A|B) is 0.3
    Full step-by-step solution

    Step 1: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Step 2: Substitute the given values: P(A|B) = 0.18 / 0.6 Step 3: Perform the division: 0.18 ÷ 0.6 = 0.3 Step 4: The conditional probability P(A|B) is 0.3

  2. log₂(16) = ? Answer: 4 Solution: We are solving: log base 2 of 16. Understand what the logarithm means. log base 2 of 16 means: "2 raised to what power equals 16?" So we write: 2^x = 16.
    Full step-by-step solution

    We are solving: log base 2 of 16. Step 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. Step 2: Express 16 as a power of 2. 16 = 2 * 2 * 2 * 2 = 2^4. Step 3: Substitute 16 with 2^4 in the equation. 2^x = 2^4. Step 4: Since the bases are the same (base 2), the exponents must be equal. Therefore, x = 4. Step 5: Conclusion. log base 2 of 16 = 4. Final answer: 4

  3. P(A|B) = P(A∩B) / P(B) where P(A) = 0.7, P(B) = 0.8, P(A∩B) = 0.56 = ? Answer: 0.7 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.56 / 0.8 Perform the division: 0.56 ÷ 0.8 = 0.7 The conditional probability P(A|B) is 0.7
    Full step-by-step solution

    Step 1: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Step 2: Substitute the given values: P(A|B) = 0.56 / 0.8 Step 3: Perform the division: 0.56 ÷ 0.8 = 0.7 Step 4: The conditional probability P(A|B) is 0.7

  4. Mere runs a diagnostic clinic that tests for a rare genetic condition. In the general population, 2% of people have the condition. The test has a 96% true positive rate (correctly identifies those with the condition) and a 4% false positive rate (incorrectly flags those without the condition). If a randomly selected person from the population tests positive for the condition, what is the probability that they actually have the condition? Express your answer as a percentage rounded to two decimal places. Answer: 32.88% Solution: Define the events. Let C = has the condition, T = tests positive. Write down the given probabilities.
    Full step-by-step solution

    Step 1: Define the events. Let C = has the condition, T = tests positive. Step 2: Write down the given probabilities. P(C) = 0.02 (2% have the condition) P(not C) = 1 - 0.02 = 0.98 P(T|C) = 0.96 (96% true positive rate) P(T|not C) = 0.04 (4% false positive rate) Step 3: We want P(C|T). Use Bayes' theorem: P(C|T) = P(T|C) * P(C) / P(T) Step 4: Find P(T) using the law of total probability. P(T) = P(T|C)*P(C) + P(T|not C)*P(not C) P(T) = (0.96 * 0.02) + (0.04 * 0.98) P(T) = 0.0192 + 0.0392 P(T) = 0.0584 Step 5: Apply Bayes' theorem. P(C|T) = (0.96 * 0.02) / 0.0584 P(C|T) = 0.0192 / 0.0584 P(C|T) = 0.328767... Step 6: Convert to percentage. 0.328767 * 100 = 32.8767% ≈ 32.88% The answer is 32.88%.

  5. P(A) = 0.81, P(B) = 0.36, P(A∩B) = 0.2916. Find P(A|B). Answer: 0.81 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.2916 / 0.36 Perform the division: 0.2916 ÷ 0.36 = 0.81 The conditional probability P(A|B) is 0.81.
    Full step-by-step solution

    Step 1: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Step 2: Substitute the given values: P(A|B) = 0.2916 / 0.36 Step 3: Perform the division: 0.2916 ÷ 0.36 = 0.81 Step 4: The conditional probability P(A|B) is 0.81.

  6. Matiu runs a community garden. He surveyed 240 members to find out their gardening preferences. He found that 144 members grow vegetables, 108 members grow flowers, and 72 members grow both vegetables and flowers. If a randomly selected member from the survey grows vegetables, what is the probability that they also grow flowers? Express your answer as a simplified fraction. Answer: 1/2 Solution: Define events. Let V be the event that a member grows vegetables, and F be the event that a member grows flowers. Use the conditional probability formula: P(F|V) = P(F and V) / P(V).
    Full step-by-step solution

    Step 1: Define events. Let V be the event that a member grows vegetables, and F be the event that a member grows flowers. We want P(F|V). Step 2: Use the conditional probability formula: P(F|V) = P(F and V) / P(V). Step 3: Find P(F and V). The number of members who grow both vegetables and flowers is 72. Total members is 240. So P(F and V) = 72/240. Step 4: Simplify P(F and V) = 72/240 = 3/10 (dividing numerator and denominator by 24). Step 5: Find P(V). The number of members who grow vegetables is 144. So P(V) = 144/240. Step 6: Simplify P(V) = 144/240 = 3/5 (dividing numerator and denominator by 48). Step 7: Apply the formula: P(F|V) = (3/10) / (3/5) = (3/10) * (5/3) = 15/30 = 1/2. The answer is 1/2.

  7. Isabella volunteers at a local animal shelter. She recorded data on 200 adopted pets. Of these, 120 were dogs and 80 were cats. Among the dogs, 72 were adopted by families with children. Among the cats, 27 were adopted by families with children. If a randomly selected adopted pet was adopted by a family with children, what is the probability that it was a dog? Express your answer as a simplified fraction. Answer: 8/11 Solution: Find the number of pets adopted by families with children. Dogs with children: 72 Cats with children: 27 Total with children: 72 + 27 = 99 We want P(dog | children).
    Full step-by-step solution

    Step 1: Find the number of pets adopted by families with children. Dogs with children: 72 Cats with children: 27 Total with children: 72 + 27 = 99 Step 2: We want P(dog | children). This is the probability that the pet is a dog, given it was adopted by a family with children. Step 3: Use the conditional probability formula: P(A|B) = P(A and B) / P(B) Here, A = dog, B = adopted by family with children. Step 4: P(dog and children) = 72/200 P(children) = 99/200 Step 5: P(dog | children) = (72/200) / (99/200) = 72/99 Step 6: Simplify the fraction. Divide numerator and denominator by 9: 72/99 = 8/11. The answer is 8/11.