Conditional Probability

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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) = 0.9, P(B) = 0.8, P(A∩B) = 0.72 = ? Answer: 0.9 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.72 / 0.8 Perform the division: 0.72 ÷ 0.8 = 0.9 The conditional probability P(A|B) is 0.9
   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) = 0.72 / 0.8 Step 3: Perform the division: 0.72 ÷ 0.8 = 0.9 Step 4: The conditional probability P(A|B) is 0.9

2. Mason runs a factory that produces electronic components. On a given day, 12% of the components are defective. The quality control system correctly identifies 95% of defective components as defective, but it also incorrectly flags 8% of non-defective components as defective. If a randomly selected component is flagged as defective by the quality control system, what is the probability that it is actually defective? Express your answer as a percentage rounded to two decimal places. Answer: 61.82% Solution: Define the events. Let D be the event that a component is defective. We are given: P(D) = 0.12, P(F|D) = 0.95, P(F|not D) = 0.08.
   Full step-by-step solution

   Step 1: Define the events. Let D be the event that a component is defective. Let F be the event that the component is flagged as defective. We are given: P(D) = 0.12, P(F|D) = 0.95, P(F|not D) = 0.08. We want P(D|F). Step 2: Find P(F and D), the probability a component is defective and flagged. P(F and D) = P(F|D) * P(D) = 0.95 * 0.12 = 0.114. Step 3: Find P(not D), the probability a component is not defective. P(not D) = 1 - P(D) = 1 - 0.12 = 0.88. Step 4: Find P(F and not D), the probability a component is not defective but flagged. P(F and not D) = P(F|not D) * P(not D) = 0.08 * 0.88 = 0.0704. Step 5: Find P(F), the total probability a component is flagged. P(F) = P(F and D) + P(F and not D) = 0.114 + 0.0704 = 0.1844. Step 6: Apply the conditional probability formula. P(D|F) = P(F and D) / P(F) = 0.114 / 0.1844 = 0.61822... Step 7: Convert to a percentage rounded to two decimal places: 0.61822 * 100% = 61.82%. The answer is 61.82%.

3. P(A|B) = P(A∩B) / P(B) where P(A) = 0.4, P(B) = 0.5, and P(A∩B) = 0.2 = ? Answer: 0.4 Solution: P(A|B) = P(A∩B) / P(B) P(A) = 0.4 P(B) = 0.5 P(A∩B) = 0.2 Identify what is needed. The problem asks for P(A|B), so we use the formula directly. Substitute the known numbers into the formula.
   Full step-by-step solution

   We are given the conditional probability formula: P(A|B) = P(A∩B) / P(B) Given values: P(A) = 0.4 P(B) = 0.5 P(A∩B) = 0.2 Step 1: Identify what is needed. The problem asks for P(A|B), so we use the formula directly. Step 2: Substitute the known numbers into the formula. P(A|B) = 0.2 / 0.5 Step 3: Perform the division. 0.2 divided by 0.5 = 2/5 = 0.4 Step 4: Interpret the result. P(A|B) = 0.4 means that if event B has occurred, the probability of event A occurring is 0.4. Final answer: 0.4

4. Mere has a square dartboard with side length 40 cm. The dartboard is divided into two regions: a central square region painted red, and the remaining border region painted blue. The central red square is inscribed in the board such that its vertices lie at the midpoints of the sides of the outer square. If a dart thrown at the board lands at a random point on the board, what is the probability that it lands in the blue region, given that it did NOT land in the red region? Express your answer as a simplified fraction. Answer: 1 Solution: The outer square has side length 40 cm. The central red square has its vertices at the midpoints of the outer square's sides. So each side of the red square connects two midpoints, forming a 45-degree angle.
   Full step-by-step solution

   Step 1: Understand the geometry. The outer square has side length 40 cm. The central red square has its vertices at the midpoints of the outer square's sides. So each side of the red square connects two midpoints, forming a 45-degree angle. The side length of the red square can be found using the Pythagorean theorem: each half-side of the outer square is 20 cm, so the red square's side = sqrt(20^2 + 20^2) = sqrt(400 + 400) = sqrt(800) = 20*sqrt(2) cm. Step 2: Calculate the area of the outer square: Area_outer = 40 * 40 = 1600 cm^2. Step 3: Calculate the area of the red square: Area_red = (20*sqrt(2))^2 = 400 * 2 = 800 cm^2. Step 4: The blue region is the border, so Area_blue = Area_outer - Area_red = 1600 - 800 = 800 cm^2. Step 5: Define events: R = dart lands in red region, B = dart lands in blue region. Since the board is entirely covered by red and blue, P(B) = 800/1600 = 1/2, P(R) = 800/1600 = 1/2. Step 6: We need P(B | not R). The event 'not R' means the dart lands anywhere except the red region, which is exactly the blue region. So P(B and not R) = P(B) because if it lands in blue, it is automatically not in red. Therefore P(B and not R) = 800/1600 = 1/2. Step 7: P(not R) = 1 - P(R) = 1 - 1/2 = 1/2. Step 8: Apply conditional probability formula: P(B | not R) = P(B and not R) / P(not R) = (1/2) / (1/2) = 1. Step 9: Interpretation: Given that the dart did not land in the red region, it must have landed in the blue region with certainty. The probability is 1. The answer is 1.

5. Matiu runs a community health clinic. He collected data from 450 patients and found that 180 patients have high blood pressure, 135 patients have high cholesterol, and 54 patients have both conditions. If a patient is randomly selected from those who have high blood pressure, what is the probability that they also have high cholesterol? Express your answer as a simplified fraction. Answer: 3/10 Solution: Identify the events. Let A be the event that a patient has high blood pressure. Use the conditional probability formula: P(B|A) = P(A and B) / P(A).
   Full step-by-step solution

   Step 1: Identify the events. Let A be the event that a patient has high blood pressure. Let B be the event that a patient has high cholesterol. We want P(B|A), the probability that a patient has high cholesterol given they have high blood pressure. Step 2: Use the conditional probability formula: P(B|A) = P(A and B) / P(A). Step 3: Calculate P(A and B) = number of patients with both conditions / total patients = 54 / 450. Step 4: Simplify 54/450. Both numbers are divisible by 18: 54/18 = 3, 450/18 = 25. So P(A and B) = 3/25. Step 5: Calculate P(A) = number of patients with high blood pressure / total patients = 180 / 450. Step 6: Simplify 180/450. Both numbers are divisible by 90: 180/90 = 2, 450/90 = 5. So P(A) = 2/5. Step 7: Apply the formula: P(B|A) = (3/25) / (2/5) = (3/25) * (5/2) = (3 * 5) / (25 * 2) = 15/50. Step 8: Simplify 15/50. Both numbers are divisible by 5: 15/5 = 3, 50/5 = 10. So P(B|A) = 3/10. The answer is 3/10.