Conditional Probability

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Conditional Probability · Grade 8 · Worksheet 3

1. A school survey found that 60% of students play sports and 40% participate in music. If 24% of students do both activities, what percentage of students play sports but do not participate in music? Answer: 36% Solution: - Percentage of students who play sports = 60% - Percentage of students who participate in music = 40% - Percentage of students who do both = 24% We want the percentage of students who play sports but do not participate in music.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the given data** - Percentage of students who play sports = 60% - Percentage of students who participate in music = 40% - Percentage of students who do both = 24% --- **Step 2: Identify what is being asked** We want the percentage of students who play sports but do **not** participate in music. That means: We take all sports players and subtract those who also do music. --- **Step 3: Use a set reasoning approach** Let S = students who play sports Let M = students who participate in music We know: S = 60% M = 40% S and M both = 24% The students who play sports but not in music = S - (S and M) --- **Step 4: Perform the calculation** Sports but not music = 60% - 24% = 36% --- **Step 5: Verify the answer** Check: Sports only = 36% Both = 24% So total sports = 36% + 24% = 60% ✔ Music only = 40% - 24% = 16% Neither = 100% - (36% + 24% + 16%) = 24% — possible, no contradiction. --- **Final Answer:** 36%

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

   Step 1: Total pencils = 8 red + 4 blue = 12 pencils Step 2: Probability first pencil is red = 8/12 = 2/3 Step 3: After drawing one red pencil, we have 7 red pencils left and total of 11 pencils Step 4: Probability second pencil is red = 7/11 Step 5: Multiply the probabilities: (8/12) × (7/11) = (2/3) × (7/11) = 14/33 The answer is 14/33.

3. A bag contains 5 red marbles, 4 blue marbles, and 3 green marbles. If two marbles are drawn without replacement, what is the probability that both are red? Answer: 5/33 Solution: Calculate total marbles: 5 red + 4 blue + 3 green = 12 marbles Probability first marble is red: 5/12 After drawing one red marble, there are 4 red marbles left out of 11 total marbles Probability second marble is red: 4/11 Multiply the probabilities: (5/12) × (4/11) = 20/132 Simplify the…
   Full step-by-step solution

   Step 1: Calculate total marbles: 5 red + 4 blue + 3 green = 12 marbles Step 2: Probability first marble is red: 5/12 Step 3: After drawing one red marble, there are 4 red marbles left out of 11 total marbles Step 4: Probability second marble is red: 4/11 Step 5: Multiply the probabilities: (5/12) × (4/11) = 20/132 Step 6: Simplify the fraction: 20/132 = 5/33 The answer is 5/33.

4. A school survey found that 60% of students participate in sports and 40% participate in music. If 24% of students participate in both sports and music, are these two activities independent? Show your reasoning. Answer: B. no Solution: Two events are independent if the probability of both occurring equals the product of their individual probabilities. This means knowing one event occurred doesn't change the likelihood of the other occurring.
   Full step-by-step solution

   Two events are independent if the probability of both occurring equals the product of their individual probabilities. This means knowing one event occurred doesn't change the likelihood of the other occurring. In real-world contexts, this helps us understand whether different activities or characteristics are related to each other.

5. A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. If two marbles are drawn without replacement, what is the probability that both are red? Answer: 1/11 Solution: Determine the total number of marbles initially. Red marbles = 4 Blue marbles = 3 Green marbles = 5 Total marbles = 4 + 3 + 5 = 12 Understand the problem.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Determine the total number of marbles initially.** Red marbles = 4 Blue marbles = 3 Green marbles = 5 Total marbles = 4 + 3 + 5 = 12 --- **Step 2: Understand the problem.** We are drawing two marbles without replacement, and we want the probability that both are red. Without replacement means the total number of marbles decreases after the first draw. --- **Step 3: Probability that the first marble is red.** Number of red marbles = 4 Total marbles = 12 P(first red) = 4/12 = 1/3 --- **Step 4: Probability that the second marble is red given the first was red.** After drawing 1 red marble: Red marbles left = 3 Total marbles left = 11 P(second red | first red) = 3/11 --- **Step 5: Multiply the probabilities for both events.** P(both red) = P(first red) × P(second red | first red) = (1/3) × (3/11) = 3/33 = 1/11 --- **Final Answer:** 1/11

6. Ava draws a 4x4 grid of squares, 16 squares total. She colors 6 squares blue and the rest red. Then, she draws a circle inside each blue square and a triangle inside each red square. If you randomly pick one square, what is the probability that it has a triangle, given that it is NOT blue? Are the events 'the square has a triangle' and 'the square is not blue' independent? Explain your answer. Answer: 1; Yes, they are independent. Solution: Define the events. Let T be the event 'the square has a triangle'. There are 16 total squares.
   Full step-by-step solution

   Step 1: Define the events. Let T be the event 'the square has a triangle'. Let B be the event 'the square is blue'. The problem asks for P(T | not B). Step 2: Calculate P(T | not B). There are 16 total squares. 6 are blue, so 16 - 6 = 10 are not blue (red). All red squares have a triangle. So, out of the 10 squares that are not blue, all 10 have a triangle. P(T | not B) = (Number of squares that are not blue AND have a triangle) / (Number of squares that are not blue) = 10 / 10 = 1. Step 3: Check for independence. Events A and B are independent if P(A and B) = P(A) * P(B). Here, we check if P(T and not B) = P(T) * P(not B). P(T) = (Number of squares with a triangle) / (Total squares). All red squares (10) have a triangle, so P(T) = 10 / 16 = 5/8. P(not B) = (Number of squares that are not blue) / (Total squares) = 10 / 16 = 5/8. P(T and not B) = (Number of squares that are not blue AND have a triangle) / (Total squares) = 10 / 16 = 5/8. Step 4: Compare. P(T) * P(not B) = (5/8) * (5/8) = 25/64. P(T and not B) = 5/8 = 40/64. Since 25/64 does not equal 40/64, the events are NOT independent. (Wait, the calculation shows P(T|not B) = 1, and for the answer I stated 1; Yes they are independent. Let me re-check the independence condition carefully.) Re-check: The condition for independence is P(A|B) = P(A). Here, P(T | not B) = 1. P(T) = 10/16 = 5/8. Since 1 is not equal to 5/8, the events are dependent. My initial answer was wrong. Corrected Step 4: P(T | not B) = 1. P(T) = 5/8. Since P(T | not B) != P(T), the events are dependent. Corrected Final Answer: The probability is 1. The events are not independent.

7. Liam is analyzing weather data for his science project. He found that on 40% of days it rains, and on 30% of days it is windy. If rain and wind are independent weather events, what percentage of days would you expect to be both rainy and windy? Answer: 12 Solution: Probability of rain = 40% = 0.40 Probability of wind = 30% = 0.30 Since the events are independent, multiply the probabilities Probability of both rain and wind = 0.40 × 0.30 0.40 × 0.30 = 0.12 0.12 = 12% The answer is 12.
   Full step-by-step solution

   Step 1: Identify the given probabilities Probability of rain = 40% = 0.40 Probability of wind = 30% = 0.30 Step 2: Since the events are independent, multiply the probabilities Probability of both rain and wind = 0.40 × 0.30 Step 3: Calculate the product 0.40 × 0.30 = 0.12 Step 4: Convert back to percentage 0.12 = 12% The answer is 12.

8. At a middle school science fair, 48% of projects were about environmental science and 30% were about robotics. If being an environmental science project and being a robotics project are independent events, what percentage of projects would you expect to be about both environmental science and robotics? Answer: 14.4 Solution: - Probability of environmental science project: 48% = 0.48 - Probability of robotics project: 30% = 0.30 Since the events are independent, multiply the probabilities 0.48 × 0.30 = 0.144 0.144 × 100 = 14.4% The answer is 14.4.
   Full step-by-step solution

   Step 1: Identify the given probabilities - Probability of environmental science project: 48% = 0.48 - Probability of robotics project: 30% = 0.30 Step 2: Since the events are independent, multiply the probabilities 0.48 × 0.30 = 0.144 Step 3: Convert back to percentage 0.144 × 100 = 14.4% The answer is 14.4.