Conditional Probability

lessonbunny.com

Answer Key & Explanations

Conditional Probability · Grade 8 · Worksheet 2

1. A rectangular prism has dimensions of 2.5 cm by 4 cm by 6 cm. If you double all three dimensions to create a larger prism, what is the ratio of the surface area of the larger prism to the surface area of the original prism? Answer: 4 Solution: The formula for the surface area \( SA \) of a rectangular prism with dimensions \( l \times w \times h \) is: SA = 2(lw + lh + wh) Original dimensions: \( l = 2.5 \), \( w = 4 \), \( h = 6 \) (all in cm).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the surface area of the original prism** The formula for the surface area \( SA \) of a rectangular prism with dimensions \( l \times w \times h \) is: \[ SA = 2(lw + lh + wh) \] Original dimensions: \( l = 2.5 \), \( w = 4 \), \( h = 6 \) (all in cm). First compute each face area sum: \[ lw = 2.5 \times 4 = 10 \] \[ lh = 2.5 \times 6 = 15 \] \[ wh = 4 \times 6 = 24 \] Sum: \( 10 + 15 + 24 = 49 \) Multiply by 2: \( SA_{\text{original}} = 2 \times 49 = 98 \) cm². --- **Step 2: Find the surface area of the larger prism** Doubled dimensions: \( l' = 5 \), \( w' = 8 \), \( h' = 12 \). \[ l'w' = 5 \times 8 = 40 \] \[ l'h' = 5 \times 12 = 60 \] \[ w'h' = 8 \times 12 = 96 \] Sum: \( 40 + 60 + 96 = 196 \) Multiply by 2: \( SA_{\text{new}} = 2 \times 196 = 392 \) cm². --- **Step 3: Ratio of new surface area to original surface area** \[ \text{Ratio} = \frac{392}{98} = 4 \] --- **Step 4: Explanation of the shortcut** When all dimensions are doubled, each face area changes by a factor of \( 2 \times 2 = 4 \) because both length and width of that face are doubled. Since surface area is the sum of the areas of all faces, and each face area is multiplied by 4, the total surface area is also multiplied by 4. Thus, the ratio is \( 4 \) without needing to compute exact numbers. --- **Final Answer:** 4

2. A rectangular prism is drawn on a coordinate plane with vertices at (0,0,0), (4,0,0), (4,3,0), (0,3,0), (0,0,2), (4,0,2), (4,3,2), and (0,3,2). A smaller prism is created by translating the original prism 1 unit in the positive x-direction and 1 unit in the positive y-direction. What is the volume of the space where the two prisms overlap? Answer: 12 Solution: Original prism dimensions: length = 4, width = 3, height = 2 Translation moves prism by +1 in x-direction and +1 in y-direction Original prism occupies x from 0 to 4, y from 0 to 3, z from 0 to 2 Translated prism occupies x from 1 to 5, y from 1 to 4, z from 0 to 2 Overlap in x-direction: from…
   Full step-by-step solution

   Step 1: Original prism dimensions: length = 4, width = 3, height = 2 Step 2: Translation moves prism by +1 in x-direction and +1 in y-direction Step 3: Original prism occupies x from 0 to 4, y from 0 to 3, z from 0 to 2 Step 4: Translated prism occupies x from 1 to 5, y from 1 to 4, z from 0 to 2 Step 5: Overlap in x-direction: from max(0,1) to min(4,5) = 1 to 4, length = 3 Step 6: Overlap in y-direction: from max(0,1) to min(3,4) = 1 to 3, width = 2 Step 7: Overlap in z-direction: from max(0,0) to min(2,2) = 0 to 2, height = 2 Step 8: Volume of overlap = length × width × height = 3 × 2 × 2 = 12 The answer is 12.

3. A weather station reports that there is a 40% chance of rain on Saturday and a 30% chance of rain on Sunday. If the probability of rain on Sunday is independent of whether it rains on Saturday, what is the probability that it rains on at least one of the two days? Answer: 0.58 Solution: Find the probability that it does NOT rain on Saturday: 1 - 0.40 = 0.60 Find the probability that it does NOT rain on Sunday: 1 - 0.30 = 0.70 Since the events are independent, multiply to find the probability that it does NOT rain on both days: 0.60 × 0.70 = 0.42 The probability that it rains on…
   Full step-by-step solution

   Step 1: Find the probability that it does NOT rain on Saturday: 1 - 0.40 = 0.60 Step 2: Find the probability that it does NOT rain on Sunday: 1 - 0.30 = 0.70 Step 3: Since the events are independent, multiply to find the probability that it does NOT rain on both days: 0.60 × 0.70 = 0.42 Step 4: The probability that it rains on at least one day is 1 minus the probability of no rain on both days: 1 - 0.42 = 0.58 The answer is 0.58.

4. Tane has a spinner divided into 5 equal sections numbered 1, 3, 5, 7, and 9. He also has a bag containing 7 marbles: three red marbles, two blue marbles, and two green marbles. He spins the spinner once and then draws one marble from the bag without looking. Let event A be that the spinner lands on an odd number greater than 5, and event B be that he draws a red marble. Are events A and B independent? Show your reasoning using the conditional probability formula. Answer: Yes, events A and B are independent Solution: Find P(A). The spinner has 5 equal sections: 1, 3, 5, 7, 9. Odd numbers greater than 5 are 7 and 9.
   Full step-by-step solution

   Step 1: Find P(A). The spinner has 5 equal sections: 1, 3, 5, 7, 9. Odd numbers greater than 5 are 7 and 9. So P(A) = 2/5. Step 2: Find P(B). The bag has 7 marbles: 3 red, 2 blue, 2 green. So P(B) = 3/7. Step 3: Find P(A and B). For event A and B to both happen, the spinner must land on 7 or 9 (2 outcomes) AND a red marble must be drawn. The probability of landing on 7 or 9 is 2/5. Given that, the probability of drawing a red marble is still 3/7 because the marble draw is independent of the spin. So P(A and B) = (2/5) * (3/7) = 6/35. Step 4: Check independence using P(A) * P(B) = P(A and B). P(A) * P(B) = (2/5) * (3/7) = 6/35. Since 6/35 equals P(A and B), the events are independent. Alternatively, check P(B|A) = P(B). P(B|A) = P(A and B) / P(A) = (6/35) / (2/5) = (6/35) * (5/2) = 30/70 = 3/7 = P(B). Since P(B|A) = P(B), the events are independent. The answer is Yes, events A and B are independent.

5. A bag contains 5 red marbles, 3 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: 9/100 Solution: Find the total number of marbles. Red = 5, Blue = 3, Green = 2. Total marbles = 5 + 3 + 2 = 10.
   Full step-by-step solution

   Let's solve this step-by-step. Step 1: Find the total number of marbles. Red = 5, Blue = 3, Green = 2. Total marbles = 5 + 3 + 2 = 10. Step 2: Find the probability of drawing a blue marble in one draw. Number of blue marbles = 3. Probability of blue = 3/10. Step 3: Understand the replacement rule. Since the marble is replaced after the first draw, the bag still has 10 marbles for the second draw, with 3 blue marbles again. So the probability of blue on the second draw is also 3/10. Step 4: Find the probability of both events happening. When two independent events both must occur, we multiply their probabilities. Probability (both blue) = (3/10) * (3/10) = 9/100. Step 5: Conclusion. The probability that both marbles drawn are blue is 9/100.

6. A bag contains 7 red marbles, 9 blue marbles, and 5 green marbles. Two marbles are drawn at random without replacement. Are the events 'first marble is red' and 'second marble is red' independent? Justify your answer by comparing P(second is red | first is red) and P(second is red). Answer: No, they are not independent because P(second is red | first is red) = 6/20 = 3/10 and P(second is red) = 7/21 = 1/3, and 3/10 ≠ 1/3. Solution: Find P(second is red) without any condition. Total marbles = 7 + 9 + 5 = 21. P(second is red) = number of red marbles / total marbles = 7/21 = 1/3.
   Full step-by-step solution

   Step 1: Find P(second is red) without any condition. Total marbles = 7 + 9 + 5 = 21. P(second is red) = number of red marbles / total marbles = 7/21 = 1/3. Step 2: Find P(second is red | first is red). If the first marble is red, then there are 6 red marbles left and 20 total marbles remaining. So P(second is red | first is red) = 6/20 = 3/10. Step 3: Compare the two probabilities. P(second is red | first is red) = 3/10 = 0.3, and P(second is red) = 1/3 ≈ 0.333. Since 3/10 ≠ 1/3, the events are not independent. The answer is: No, they are not independent.