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 bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. If you randomly draw one marble, replace it, and then draw a second marble, what is the probability that both marbles drawn are blue? Answer: 1/4 Solution: Find the total number of marbles. 3 red + 5 blue + 2 green = 10 marbles. Find the probability of drawing a blue marble on the first draw.
   Full step-by-step solution

   Step 1: Find the total number of marbles. 3 red + 5 blue + 2 green = 10 marbles. Step 2: Find the probability of drawing a blue marble on the first draw. P(blue first) = Number of blue marbles / Total marbles = 5/10 = 1/2. Step 3: Because the marble is replaced, the total number of marbles is still 10 for the second draw. The probability of drawing a blue marble on the second draw is also P(blue second) = 5/10 = 1/2. Step 4: Since the draws are independent (due to replacement), multiply the probabilities. P(both blue) = P(blue first) × P(blue second) = (1/2) × (1/2) = 1/4. The answer is 1/4.

3. Sophia rolls a fair 6-sided die and flips a fair coin. Event A is rolling an even number. Event B is flipping heads. Are events A and B independent? Answer: Yes Solution: Find P(A). The die has 6 sides, even numbers are 2, 4, 6. So P(A) = 3/6 = 1/2.
   Full step-by-step solution

   Step 1: Find P(A). The die has 6 sides, even numbers are 2, 4, 6. So P(A) = 3/6 = 1/2. Step 2: Find P(B). The coin has 2 sides, heads is 1 outcome. So P(B) = 1/2. Step 3: Find P(A and B). The sample space has 6 × 2 = 12 equally likely outcomes. Outcomes where die is even and coin is heads: (2,H), (4,H), (6,H) → 3 outcomes. So P(A and B) = 3/12 = 1/4. Step 4: Multiply P(A) × P(B) = (1/2) × (1/2) = 1/4. Step 5: Since P(A and B) = 1/4 = P(A) × P(B), the events are independent. The answer is Yes.

4. A school survey found that 60% of students play sports, and 25% of students play a musical instrument. Among those who play sports, 30% also play a musical instrument. What percentage of students play a musical instrument but do not play sports? Answer: 7% Solution: - Let total students = 100% (for easier calculation). - Students who play sports = 60% - Students who play a musical instrument = 25% - Among those who play sports, 30% also play a musical instrument.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define the given percentages** - Let total students = 100% (for easier calculation). - Students who play sports = 60% - Students who play a musical instrument = 25% - Among those who play sports, 30% also play a musical instrument. --- **Step 2: Find students who play both sports and a musical instrument** 30% of sports players play a musical instrument. Sports players = 60% So: Both sports and music = 30% of 60% = (30/100) * 60 = 0.30 * 60 = 18% of all students. --- **Step 3: Find students who play music but not sports** Total music players = 25% Music players who also play sports = 18% So: Music but not sports = Total music players − Both = 25% − 18% = 7% --- **Step 4: Conclusion** The percentage of students who play a musical instrument but do not play sports is **7%**.

5. Sophia has a bag containing 6 red marbles and 4 blue marbles. She draws one marble, records its color, and then replaces it. Then she draws a second marble. A visual diagram shows a tree with two stages: first draw branches into 'Red' (6/10) and 'Blue' (4/10), and from each of those, second draw branches again into 'Red' and 'Blue' with the same probabilities. What is the probability that the second marble is red, given that the first marble was blue? Are the events 'first marble is blue' and 'second marble is red' independent? Answer: 0.6, Yes they are independent Solution: Total marbles = 6 red + 4 blue = 10. Since the marble is replaced, the probability of drawing a red marble on any draw is always 6/10 = 3/5 = 0.6.
   Full step-by-step solution

   Step 1: Total marbles = 6 red + 4 blue = 10. Since the marble is replaced, the probability of drawing a red marble on any draw is always 6/10 = 3/5 = 0.6. Similarly, probability of blue is 4/10 = 2/5 = 0.4. Step 2: P(second marble is red given first marble is blue) = P(second is red) because of replacement — the first draw does not affect the second. So P(second red | first blue) = 6/10 = 3/5 = 0.6. Step 3: To check independence, we compare P(first blue and second red) with P(first blue) * P(second red). P(first blue) = 4/10 = 0.4 P(second red) = 6/10 = 0.6 P(first blue and second red) = (4/10)*(6/10) = 24/100 = 0.24 P(first blue) * P(second red) = 0.4 * 0.6 = 0.24 Since these are equal, the events are independent. The answer is 0.6, Yes they are independent.

6. Olivia surveyed 50 students at her school to determine their favorite after-school activities. She found that 20 students enjoy reading, and 15 students enjoy playing sports. If the events 'enjoys reading' and 'enjoys playing sports' are independent, how many students would Olivia expect to enjoy both reading and playing sports? Answer: 6 Solution: Find the probability a student enjoys reading: P(reading) = 20/50 = 0.4 Find the probability a student enjoys playing sports: P(sports) = 15/50 = 0.3 For independent events, multiply the probabilities: P(reading and sports) = 0.4 * 0.3 = 0.12 Convert the probability to an expected number of…
   Full step-by-step solution

   Step 1: Find the probability a student enjoys reading: P(reading) = 20/50 = 0.4 Step 2: Find the probability a student enjoys playing sports: P(sports) = 15/50 = 0.3 Step 3: For independent events, multiply the probabilities: P(reading and sports) = 0.4 * 0.3 = 0.12 Step 4: Convert the probability to an expected number of students: 0.12 * 50 = 6 So Olivia would expect 6 students to enjoy both reading and playing sports.

7. At a community center, 70% of members participate in fitness classes and 40% participate in art workshops. If participation in fitness classes and art workshops are independent events, what percentage of members participate in both activities? Answer: 28 Solution: - Probability of fitness class participation = 70% = 0.70 - Probability of art workshop participation = 40% = 0.40 Since the events are independent, multiply the probabilities Probability(both) = 0.70 × 0.40 0.70 × 0.40 = 0.28 0.28 = 28% The answer is 28.
   Full step-by-step solution

   Step 1: Identify the given probabilities - Probability of fitness class participation = 70% = 0.70 - Probability of art workshop participation = 40% = 0.40 Step 2: Since the events are independent, multiply the probabilities Probability(both) = 0.70 × 0.40 Step 3: Calculate the product 0.70 × 0.40 = 0.28 Step 4: Convert back to percentage 0.28 = 28% The answer is 28.