Conditional Probability

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Answer Key & Explanations

Conditional Probability · Grade 8 · Worksheet 3

1. P(A) = 1/6, P(B) = 1/6, P(A and B) = 1/36. Are events A and B independent? Answer: Yes Solution: Calculate P(A) * P(B) = (1/6) * (1/6) = 1/36. Compare with P(A and B) = 1/36. Since P(A) * P(B) = P(A and B), the events are independent.
   Full step-by-step solution

   Step 1: Calculate P(A) * P(B) = (1/6) * (1/6) = 1/36. Step 2: Compare with P(A and B) = 1/36. Step 3: Since P(A) * P(B) = P(A and B), the events are independent. The answer is Yes.

2. 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%

3. 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.

4. Mason surveyed 72 students at his school to find out their favorite subjects. He found that 42% of students prefer mathematics and 27% of students prefer science. If preferring mathematics and preferring science are independent events, what percentage of students would you expect to prefer both mathematics and science? Answer: 11.34 Solution: Convert the percentages to decimals. 42% = 0.42, 27% = 0.27. For independent events, multiply the probabilities: 0.42 × 0.27 = 0.1134.
   Full step-by-step solution

   Step 1: Convert the percentages to decimals. 42% = 0.42, 27% = 0.27. Step 2: For independent events, multiply the probabilities: 0.42 × 0.27 = 0.1134. Step 3: Convert the decimal back to a percentage: 0.1134 × 100 = 11.34%. The answer is 11.34%.

5. A local animal shelter found that 40% of their dogs are adopted within the first week, and 30% of their dogs are large breeds. If dog adoption and being a large breed are independent events, what percentage of dogs at the shelter would you expect to be both adopted within the first week AND large breeds? Answer: 12 Solution: Convert percentages to decimals: 40% = 0.40, 30% = 0.30 For independent events, multiply the probabilities: 0.40 × 0.30 = 0.12 Convert back to percentage: 0.12 = 12% The answer is 12.
   Full step-by-step solution

   Step 1: Convert percentages to decimals: 40% = 0.40, 30% = 0.30 Step 2: For independent events, multiply the probabilities: 0.40 × 0.30 = 0.12 Step 3: Convert back to percentage: 0.12 = 12% The answer is 12.

6. Mason has a set of 42 colored tiles arranged in a 6-by-7 grid on a table. The tiles are either blue or yellow. Looking at the grid, you see that 12 tiles in the first two rows are blue, and 17 tiles in the entire grid are blue. If you randomly pick a tile from the first two rows, what is the probability it is blue? Then, check if the events 'the tile is from the first two rows' and 'the tile is blue' are independent by comparing P(blue) and P(blue | first two rows). Answer: No, they are not independent Solution: Total tiles = 42. Blue tiles total = 17. So P(blue) = 17/42.
   Full step-by-step solution

   Step 1: Total tiles = 42. Blue tiles total = 17. So P(blue) = 17/42. Step 2: First two rows have 2 rows x 7 tiles per row = 14 tiles. Blue tiles in first two rows = 12. So P(blue | first two rows) = 12/14 = 6/7. Step 3: Check independence: If independent, P(blue | first two rows) should equal P(blue). Compare 6/7 and 17/42. 6/7 = 36/42, which is not equal to 17/42. So the events are not independent. The answer is: No, they are not independent.

7. A scientist is studying bacteria growth. A petri dish initially contains 2.5 × 10^4 bacteria. The bacteria population doubles every hour. Write an equation in the form y = a × b^x that models this situation, where y is the total number of bacteria and x is the number of hours that have passed. Answer: y=25000*2^x Solution: Identify the initial amount of bacteria: 2.5 × 10^4 Convert scientific notation to standard form: 2.5 × 10^4 = 25,000 Identify the growth factor: the population doubles, so b = 2 Write the exponential equation: y = a × b^x Substitute the values: y = 25,000 × 2^x The equation is y = 25000 × 2^x.
   Full step-by-step solution

   Step 1: Identify the initial amount of bacteria: 2.5 × 10^4 Step 2: Convert scientific notation to standard form: 2.5 × 10^4 = 25,000 Step 3: Identify the growth factor: the population doubles, so b = 2 Step 4: Write the exponential equation: y = a × b^x Step 5: Substitute the values: y = 25,000 × 2^x The equation is y = 25000 × 2^x.