Probability Rules

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Probability Rules · Grade 10 · Worksheet 1

1. P(Matiu studies) = 0.8, P(Matiu passes | Matiu studies) = 0.6, P(Matiu passes and Matiu studies) = ? Answer: 0.48 Solution: We are given P(Matiu studies) = 0.8 and P(Matiu passes | Matiu studies) = 0.6. The multiplication rule states: P(A and B) = P(A) × P(B|A). Here, A = 'Matiu studies' and B = 'Matiu passes'.
   Full step-by-step solution

   Step 1: We are given P(Matiu studies) = 0.8 and P(Matiu passes | Matiu studies) = 0.6. Step 2: The multiplication rule states: P(A and B) = P(A) × P(B|A). Step 3: Here, A = 'Matiu studies' and B = 'Matiu passes'. Step 4: So, P(Matiu passes and Matiu studies) = P(Matiu studies) × P(Matiu passes | Matiu studies) = 0.8 × 0.6. Step 5: 0.8 × 0.6 = 0.48. The answer is 0.48.

2. P(Tane passes science) = 0.85, P(Tane passes English) = 0.90, P(Tane passes both) = 0.765. Find P(Tane passes English | Tane passes science). Answer: 0.9 Solution: Identify the events. Let A be 'Tane passes English' and B be 'Tane passes science'. We are given P(A) = 0.90, P(B) = 0.85, and P(A and B) = 0.765.
   Full step-by-step solution

   Step 1: Identify the events. Let A be 'Tane passes English' and B be 'Tane passes science'. We are given P(A) = 0.90, P(B) = 0.85, and P(A and B) = 0.765. We need to find P(A|B). Step 2: Apply the conditional probability formula: P(A|B) = P(A and B) / P(B). Step 3: Substitute the given values: P(A|B) = 0.765 / 0.85. Step 4: Perform the division: 0.765 ÷ 0.85 = 0.9. Step 5: The conditional probability P(Tane passes English | Tane passes science) is 0.9.

3. P(A) = 0.7, P(B) = 0.5, P(A and B) = 0.35. Find P(A|B). Answer: 0.7 Solution: Step 1: Write the conditional probability formula: P(A|B) = P(A and B) / P(B) Step 2: Substitute the given values: P(A|B) = 0.35 / 0.5 Step 3: Calculate the division: 0.35 ÷ 0.5 = 0.7 Step 4: Verify independence: P(A) × P(B) = 0.7 × 0.5 = 0.35, which equals P(A and B), confirming the events are…
   Full step-by-step solution

   Step 1: Write the conditional probability formula: P(A|B) = P(A and B) / P(B) Step 2: Substitute the given values: P(A|B) = 0.35 / 0.5 Step 3: Calculate the division: 0.35 ÷ 0.5 = 0.7 Step 4: Verify independence: P(A) × P(B) = 0.7 × 0.5 = 0.35, which equals P(A and B), confirming the events are independent Step 5: The answer is 0.7

4. Kaia is testing a new medical diagnostic device. The probability that a patient has a certain condition is 0.15. If a patient has the condition, the probability the device detects it is 0.91. If a patient does not have the condition, the probability the device incorrectly indicates the condition is 0.07. What is the probability that a patient actually has the condition given that the device indicates they do? Answer: A. 0.67 Solution: Let C = patient has the condition (P(C) = 0.15) Let D = device indicates condition P(C) = 0.15 P(D|C) = 0.91 (true positive rate) P(D|not C) = 0.07 (false positive rate) P(not C) = 1 - P(C) = 1 - 0.15 = 0.85 Use the multiplication rule to find P(D and C) and P(D and not C) P(D and C) = P(C) ×…
   Full step-by-step solution

   Step 1: Define the events Let C = patient has the condition (P(C) = 0.15) Let D = device indicates condition Step 2: Write the given probabilities P(C) = 0.15 P(D|C) = 0.91 (true positive rate) P(D|not C) = 0.07 (false positive rate) Step 3: Calculate P(not C) P(not C) = 1 - P(C) = 1 - 0.15 = 0.85 Step 4: Use the multiplication rule to find P(D and C) and P(D and not C) P(D and C) = P(C) × P(D|C) = 0.15 × 0.91 = 0.1365 P(D and not C) = P(not C) × P(D|not C) = 0.85 × 0.07 = 0.0595 Step 5: Find P(D) using the law of total probability P(D) = P(D and C) + P(D and not C) = 0.1365 + 0.0595 = 0.196 Step 6: Apply Bayes' theorem to find P(C|D) P(C|D) = P(D and C) / P(D) = 0.1365 / 0.196 = 0.6964 ≈ 0.70 The closest answer is 0.67.

5. P(Ava passes math) = 0.8, P(Ava passes science | Ava passes math) = 0.75. Find P(Ava passes math and science). Answer: 0.6 Solution: Identify the given probabilities. P(Ava passes math) = 0.8 P(Ava passes science | Ava passes math) = 0.75 Apply the multiplication rule for dependent events: P(A and B) = P(A) × P(B|A).
   Full step-by-step solution

   Step 1: Identify the given probabilities. P(Ava passes math) = 0.8 P(Ava passes science | Ava passes math) = 0.75 Step 2: Apply the multiplication rule for dependent events: P(A and B) = P(A) × P(B|A). Let A = 'Ava passes math' Let B = 'Ava passes science' So, P(A and B) = P(A) × P(B|A) Step 3: Substitute the given values into the formula. P(Ava passes math and science) = 0.8 × 0.75 Step 4: Perform the multiplication. 0.8 × 0.75 = 0.6 Step 5: State the final answer. The probability that Ava passes both math and science is 0.6.

6. P(Emma passes physics) = 0.85, P(Emma passes chemistry) = 0.75, P(Emma passes both) = 0.65. Find P(Emma passes chemistry | Emma passes physics). Answer: 0.7647 Solution: Let A = Emma passes chemistry, B = Emma passes physics Given: P(A) = 0.75, P(B) = 0.85, P(A and B) = 0.65 P(A|B) = P(A and B) / P(B) P(A|B) = 0.65 / 0.85 0.65 ÷ 0.85 = 65/85 = 13/17 ≈ 0.7647 The answer is 0.7647.
   Full step-by-step solution

   Step 1: Identify the events Let A = Emma passes chemistry, B = Emma passes physics Given: P(A) = 0.75, P(B) = 0.85, P(A and B) = 0.65 Step 2: Apply the conditional probability formula P(A|B) = P(A and B) / P(B) Step 3: Substitute the values P(A|B) = 0.65 / 0.85 Step 4: Calculate the result 0.65 ÷ 0.85 = 65/85 = 13/17 ≈ 0.7647 The answer is 0.7647.

7. P(Ava) = 0.75, P(Noah|Ava) = 0.80, P(Ava and Noah) = ? Answer: 0.6 Solution: Recall the multiplication rule for conditional probability: P(A and B) = P(A) × P(B|A) Substitute the given values: P(Ava and Noah) = P(Ava) × P(Noah|Ava) Calculate: P(Ava and Noah) = 0.75 × 0.80 Multiply: 0.75 × 0.80 = 0.60 The probability that both events occur is 0.60 The answer is 0.6.
   Full step-by-step solution

   Step 1: Recall the multiplication rule for conditional probability: P(A and B) = P(A) × P(B|A) Step 2: Substitute the given values: P(Ava and Noah) = P(Ava) × P(Noah|Ava) Step 3: Calculate: P(Ava and Noah) = 0.75 × 0.80 Step 4: Multiply: 0.75 × 0.80 = 0.60 Step 5: The probability that both events occur is 0.60 The answer is 0.6.

8. Mere is analyzing survey data about music preferences. In her school, 60% of students listen to pop music and 40% listen to classical music. Among those who listen to classical music, 50% also listen to pop music. What is the probability that a randomly selected student listens to pop music given that they listen to classical music? Answer: 0.5 Solution: Identify the events - Let A be 'listens to pop music' and B be 'listens to classical music'. We are given P(A) = 0.6, P(B) = 0.4, and P(A|B) = 0.5.
   Full step-by-step solution

   Step 1: Identify the events - Let A be 'listens to pop music' and B be 'listens to classical music'. Step 2: We are given P(A) = 0.6, P(B) = 0.4, and P(A|B) = 0.5. Step 3: The question asks for P(A|B), which is the probability a student listens to pop given they listen to classical. Step 4: This is directly given in the problem as 50% or 0.5. Step 5: Therefore, the probability is 0.5. The answer is 0.5.