Conditional Probability

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Conditional Probability · Grade 10 · Worksheet 2

1. 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

2. P(A|B) = P(A∩B)/P(B) where P(A) = 0.6, P(B) = 0.4, P(A∩B) = 0.3 = ? Answer: 0.75 Solution: P(A) = 0.6 P(B) = 0.4 P(A∩B) = 0.3 We want P(A|B). P(A|B) = P(A∩B) / P(B) P(A|B) = 0.3 / 0.4 0.3 divided by 0.4 is the same as 3/4. Convert 3/4 to decimal: 3/4 = 0.75 P(A|B) = 0.75
   Full step-by-step solution

   We are given: P(A) = 0.6 P(B) = 0.4 P(A∩B) = 0.3 We want P(A|B). Step 1: Recall the conditional probability formula: P(A|B) = P(A∩B) / P(B) Step 2: Substitute the known numbers into the formula: P(A|B) = 0.3 / 0.4 Step 3: Perform the division: 0.3 divided by 0.4 is the same as 3/4. Step 4: Convert 3/4 to decimal: 3/4 = 0.75 So the final answer is: P(A|B) = 0.75

3. P(A|B) = P(A∩B) / P(B) where P(A) = 0.65, P(B) = 0.52, P(A∩B) = 0.338 = ? Answer: 0.65 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.338 / 0.52 Perform the division: 0.338 ÷ 0.52 = 0.65 The conditional probability P(A|B) is 0.65
   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) = 0.338 / 0.52 Step 3: Perform the division: 0.338 ÷ 0.52 = 0.65 Step 4: The conditional probability P(A|B) is 0.65

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

5. P(A|B) = P(A∩B) / P(B) where P(A) = 0.7, P(B) = 0.9, P(A∩B) = 0.63 = ? Answer: 0.7 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.63 / 0.9 Perform the division: 0.63 ÷ 0.9 = 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) = 0.63 / 0.9 Step 3: Perform the division: 0.63 ÷ 0.9 = 0.7 Step 4: The conditional probability P(A|B) is 0.7

6. Isabella runs a community garden club at her high school. She surveyed 127 students about their gardening preferences. She found that 72 students enjoy planting vegetables, 57 students enjoy planting flowers, and 37 students enjoy planting both vegetables and flowers. If a randomly selected student from the survey enjoys planting vegetables, what is the probability that they also enjoy planting flowers? Express your answer as a simplified fraction. Answer: 37/72 Solution: Define the events. Let V be the event that a student enjoys planting vegetables. Total students surveyed: 127 Students who enjoy vegetables (V): 72 Students who enjoy flowers (F): 57 Students who enjoy both (V and F): 37 We need P(F | V), the probability a student enjoys flowers given they enjoy…
   Full step-by-step solution

   Step 1: Define the events. Let V be the event that a student enjoys planting vegetables. Let F be the event that a student enjoys planting flowers. Step 2: Identify the given values. Total students surveyed: 127 Students who enjoy vegetables (V): 72 Students who enjoy flowers (F): 57 Students who enjoy both (V and F): 37 Step 3: We need P(F | V), the probability a student enjoys flowers given they enjoy vegetables. Step 4: Use the conditional probability formula: P(F | V) = P(V and F) / P(V) Step 5: Calculate P(V and F) = 37/127 Step 6: Calculate P(V) = 72/127 Step 7: Apply the formula: P(F | V) = (37/127) / (72/127) = (37/127) * (127/72) = 37/72 The answer is 37/72.

7. P(A) = 0.88, P(B) = 0.75, P(A∩B) = 0.66. Find P(A|B). Answer: 0.88 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.66 / 0.75 Perform the division: 0.66 ÷ 0.75 = 0.88 The conditional probability P(A|B) is 0.88.
   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) = 0.66 / 0.75 Step 3: Perform the division: 0.66 ÷ 0.75 = 0.88 Step 4: The conditional probability P(A|B) is 0.88.