Conditional Probability

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Conditional Probability · Grade 8 · Worksheet 1

1. Aroha has a spinner divided into 12 equal sections numbered 1 through 12. She spins the spinner once. Event A is 'the number is a multiple of 3' (3, 6, 9, 12). Event B is 'the number is greater than 8' (9, 10, 11, 12). Aroha draws a Venn diagram to show the sample space. Based on the visual diagram, are events A and B independent? Justify your answer by calculating P(A), P(B), P(A and B), and checking the independence condition. Answer: No Solution: The sample space has 12 equally likely outcomes: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Event A (multiples of 3): {3, 6, 9, 12}. So P(A) = 4/12 = 1/3.
   Full step-by-step solution

   Step 1: The sample space has 12 equally likely outcomes: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Step 2: Event A (multiples of 3): {3, 6, 9, 12}. So P(A) = 4/12 = 1/3. Step 3: Event B (numbers greater than 8): {9, 10, 11, 12}. So P(B) = 4/12 = 1/3. Step 4: Event A and B (numbers that are both multiples of 3 AND greater than 8): {9, 12}. So P(A and B) = 2/12 = 1/6. Step 5: Check independence condition: P(A) * P(B) = (1/3) * (1/3) = 1/9. But P(A and B) = 1/6. Step 6: Since 1/9 is not equal to 1/6, P(A) * P(B) is not equal to P(A and B). Therefore, events A and B are not independent. The answer is No.

2. A school survey found that 60% of students play sports and 40% of students play musical instruments. If playing sports and playing musical instruments are independent events, what percentage of students would you expect to both play sports and play musical instruments? Answer: 24% Solution: - 60% of students play sports. - 40% of students play musical instruments. - Playing sports and playing musical instruments are independent events.
   Full step-by-step solution

   Step 1: Understand the problem We are told: - 60% of students play sports. - 40% of students play musical instruments. - Playing sports and playing musical instruments are independent events. We need to find the percentage of students who do both. Step 2: Recall the rule for independent events If two events A and B are independent, the probability of both happening is: P(A and B) = P(A) × P(B) Step 3: Convert percentages to probabilities 60% = 60/100 = 0.60 40% = 40/100 = 0.40 Step 4: Apply the multiplication rule P(sports and music) = 0.60 × 0.40 Step 5: Perform the multiplication 0.60 × 0.40 = 0.24 Step 6: Convert back to percentage 0.24 = 24/100 = 24% Step 7: Interpret the result Since the events are independent, we expect 24% of students to both play sports and play musical instruments. Final answer: 24%

3. A city survey found that 70% of households have a pet and 40% of households have a garden. The survey also showed that 28% of households have both a pet and a garden. Are having a pet and having a garden independent events for households in this city? Show your reasoning. Answer: A. yes Solution: Two events are independent if the occurrence of one does not affect the probability of the other occurring.
   Full step-by-step solution

   Two events are independent if the occurrence of one does not affect the probability of the other occurring. This can be tested by checking if the probability of both events happening equals the product of their individual probabilities. For example, if flipping a coin and rolling a die are independent, the probability of getting heads and a 6 would be (1/2) × (1/6) = 1/12.

4. A box contains 8 red balls and 4 blue balls. If two balls are drawn randomly without replacement, what is the probability that both balls are red? Answer: 14/33 Solution: Calculate total number of balls: 8 red + 4 blue = 12 balls Probability first ball is red: 8/12 = 2/3 After drawing one red ball, there are 7 red balls left out of 11 total balls Probability second ball is red: 7/11 Multiply the probabilities: (2/3) × (7/11) = 14/33 The answer is 14/33.
   Full step-by-step solution

   Step 1: Calculate total number of balls: 8 red + 4 blue = 12 balls Step 2: Probability first ball is red: 8/12 = 2/3 Step 3: After drawing one red ball, there are 7 red balls left out of 11 total balls Step 4: Probability second ball is red: 7/11 Step 5: Multiply the probabilities: (2/3) × (7/11) = 14/33 The answer is 14/33.

5. Mere has a large rectangular board divided into 8 equal sections, each painted a different color. The board is shown in a diagram with sections labeled in two rows of four. The top row has colors: Red, Blue, Green, Yellow. The bottom row has colors: Purple, Orange, Pink, Brown. A spinner arrow is placed at the center of the board. When the spinner is spun, it lands on one section at random. Let event A be 'lands on a warm color (Red, Orange, Yellow, or Pink)' and event B be 'lands on a section in the top row'. Are events A and B independent? Justify your answer using probability calculations. Answer: Yes, they are independent Solution: Total sections = 8. Event A (warm colors): Red, Orange, Yellow, Pink = 4 sections. So P(A) = 4/8 = 1/2.
   Full step-by-step solution

   Step 1: Total sections = 8. Event A (warm colors): Red, Orange, Yellow, Pink = 4 sections. So P(A) = 4/8 = 1/2. Step 2: Event B (top row): Red, Blue, Green, Yellow = 4 sections. So P(B) = 4/8 = 1/2. Step 3: Event A and B (warm colors in top row): Red and Yellow = 2 sections. So P(A and B) = 2/8 = 1/4. Step 4: Check independence: P(A) x P(B) = (1/2) x (1/2) = 1/4. This equals P(A and B) = 1/4. Step 5: Since P(A and B) = P(A) x P(B), the events are independent. The answer is Yes, they are independent.

6. P(A) = 7/12, P(B) = 9/16, P(A and B) = 21/64. Are events A and B independent? Answer: Yes Solution: Calculate P(A) × P(B) = (7/12) × (9/16) = 63/192. Simplify 63/192 by dividing numerator and denominator by 3: 63/192 = 21/64. Compare with P(A and B) = 21/64.
   Full step-by-step solution

   Step 1: Calculate P(A) × P(B) = (7/12) × (9/16) = 63/192. Step 2: Simplify 63/192 by dividing numerator and denominator by 3: 63/192 = 21/64. Step 3: Compare with P(A and B) = 21/64. Step 4: Since P(A) × P(B) = 21/64 = P(A and B), the events are independent. The answer is Yes.

7. A bag contains 8 red marbles, 5 blue marbles, and 7 green marbles. If two marbles are drawn randomly without replacement, what is the probability that both marbles are red? Answer: 0.147 Solution: Calculate the total number of marbles: 8 red + 5 blue + 7 green = 20 marbles Calculate the probability of drawing a red marble first: P(first red) = 8/20 = 2/5 After drawing one red marble, there are 7 red marbles left and 19 total marbles Calculate the probability of drawing a second red…
   Full step-by-step solution

   Step 1: Calculate the total number of marbles: 8 red + 5 blue + 7 green = 20 marbles Step 2: Calculate the probability of drawing a red marble first: P(first red) = 8/20 = 2/5 Step 3: After drawing one red marble, there are 7 red marbles left and 19 total marbles Step 4: Calculate the probability of drawing a second red marble: P(second red) = 7/19 Step 5: Multiply the probabilities: P(both red) = (8/20) × (7/19) = (2/5) × (7/19) = 14/95 Step 6: Convert to decimal: 14 ÷ 95 = 0.147368... ≈ 0.147 The answer is 0.147.