Conditional Probability

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Answer Key & Explanations

Conditional Probability · Grade 10 · Worksheet 2

1. P(A|B) = P(A∩B) / P(B) where P(A) = 0.45, P(B) = 0.75, P(A∩B) = 0.3 = ? Answer: 0.4 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.3 / 0.75 Perform the division: 0.3 ÷ 0.75 = 0.4 The conditional probability P(A|B) is 0.4
   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) = 0.3 / 0.75 Step 3: Perform the division: 0.3 ÷ 0.75 = 0.4 Step 4: The conditional probability P(A|B) is 0.4

2. A circle with center at (0,0) and radius 5 units is drawn on a coordinate plane. A point P is randomly selected from within the square region bounded by x = -5, x = 5, y = -5, and y = 5. What is the probability that point P lies inside the circle? (Use π = 3.14) Answer: 0.785 Solution: We have a circle centered at (0,0) with radius 5, so its equation is x^2 + y^2 = 25. The square region is bounded by x = -5, x = 5, y = -5, y = 5.
   Full step-by-step solution

   Step 1: Understand the problem We have a circle centered at (0,0) with radius 5, so its equation is x^2 + y^2 = 25. The square region is bounded by x = -5, x = 5, y = -5, y = 5. We want the probability that a random point inside the square is also inside the circle. Step 2: Find the area of the square The square’s side length = 5 - (-5) = 10. Area of square = side^2 = 10 * 10 = 100 square units. Step 3: Find the area of the circle Radius r = 5. Area of circle = π * r^2 = π * 25. Given π = 3.14, Area of circle = 3.14 * 25 = 78.5 square units. Step 4: Probability formula Probability = (Area of circle) / (Area of square) = 78.5 / 100. Step 5: Simplify 78.5 / 100 = 0.785. Step 6: Conclusion The probability that a randomly chosen point from the square lies inside the circle is 0.785. Final answer: 0.785

3. A circle is inscribed in a square with side length 10 cm. A smaller square is then inscribed in the circle such that its vertices lie on the circle. What is the area of the smaller square? Answer: 50 cm² Solution: We have a square of side length 10 cm. A circle is inscribed in it — that means the circle touches all four sides of the square.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the setup** We have a square of side length 10 cm. A circle is inscribed in it — that means the circle touches all four sides of the square. So the diameter of the circle = side length of the big square = 10 cm. Thus, radius \( r = 10/2 = 5 \) cm. --- **Step 2: Position of the smaller square** The smaller square is inscribed in the circle — its vertices lie on the circumference of the circle. That means the diagonal of the smaller square = diameter of the circle. --- **Step 3: Relate diagonal of smaller square to its side length** Let side length of smaller square = \( a \). Diagonal of a square = \( a \sqrt{2} \). We know: \( a \sqrt{2} = \) diameter of circle = 10 cm. So \( a \sqrt{2} = 10 \). Thus \( a = 10 / \sqrt{2} \). --- **Step 4: Simplify \( a \)** \( a = 10 / \sqrt{2} \) Multiply numerator and denominator by \( \sqrt{2} \): \( a = (10 \sqrt{2}) / 2 = 5 \sqrt{2} \) cm. --- **Step 5: Area of smaller square** Area = \( a^2 = (5 \sqrt{2})^2 = 25 \times 2 = 50 \) cm². --- **Final answer:** 50 cm²

4. P(A|B) = P(A∩B) / P(B) where P(A) = 0.7, P(B) = 0.3, P(A∩B) = 0.21 = ? Answer: 0.7 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.21 / 0.3 Perform the division: 0.21 ÷ 0.3 = 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) = 0.21 / 0.3 Step 3: Perform the division: 0.21 ÷ 0.3 = 0.7 Step 4: The conditional probability P(A|B) is 0.7

5. Olivia surveyed 315 students at her school about their preferred lunch options. She found that 189 students prefer hot lunch, 147 students prefer cold lunch, and 63 students prefer both hot and cold lunch. If a randomly selected student from the survey prefers hot lunch, what is the probability that they also prefer cold lunch? Express your answer as a simplified fraction. Answer: 1/3 Solution: Identify the events. Let H be the event that a student prefers hot lunch. Total students: 315 Students who prefer hot lunch: 189 Students who prefer both hot and cold lunch: 63 Use the conditional probability formula: P(C|H) = P(C and H) / P(H) Calculate P(C and H).
   Full step-by-step solution

   Step 1: Identify the events. Let H be the event that a student prefers hot lunch. Let C be the event that a student prefers cold lunch. We want P(C|H), the probability that a student prefers cold lunch given they prefer hot lunch. Step 2: Write down the given numbers. Total students: 315 Students who prefer hot lunch: 189 Students who prefer both hot and cold lunch: 63 Step 3: Use the conditional probability formula: P(C|H) = P(C and H) / P(H) Step 4: Calculate P(C and H). This is the probability a student prefers both: 63/315 = 1/5 (since 63 divided by 63 is 1 and 315 divided by 63 is 5). Step 5: Calculate P(H). This is the probability a student prefers hot lunch: 189/315 = 3/5 (since 189 divided by 63 is 3 and 315 divided by 63 is 5). Step 6: Apply the formula: P(C|H) = (1/5) / (3/5) = (1/5) * (5/3) = 1/3. The answer is 1/3.