Compound Probability

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Compound Probability · Grade 7 · Worksheet 2

1. Emma is conducting a probability experiment with a bag of marbles and a standard six-sided die. The bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. She randomly draws one marble from the bag and rolls the die once. What is the probability that Emma draws a blue marble AND rolls a number greater than 4? Express your answer as a simplified fraction. Answer: 1/10 Solution: Find the probability of drawing a blue marble. Total marbles = 5 red + 3 blue + 2 green = 10 marbles Blue marbles = 3 Probability of blue marble = 3/10 Find the probability of rolling a number greater than 4 on a six-sided die.
   Full step-by-step solution

   Step 1: Find the probability of drawing a blue marble. Total marbles = 5 red + 3 blue + 2 green = 10 marbles Blue marbles = 3 Probability of blue marble = 3/10 Step 2: Find the probability of rolling a number greater than 4 on a six-sided die. Numbers greater than 4: 5 and 6 Favorable outcomes = 2 Total possible outcomes = 6 Probability = 2/6 = 1/3 Step 3: Find the probability of both events occurring. Since these are independent events, multiply the probabilities: (3/10) × (1/3) = 3/30 = 1/10 The answer is 1/10.

2. Emma is conducting a probability experiment with a bag of marbles and a standard six-sided die. The bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. Emma will randomly draw one marble from the bag and roll the die once. What is the probability that she draws a blue marble AND rolls a number greater than 4? Express your answer as a simplified fraction. Answer: 1/15 Solution: Find the probability of drawing a blue marble. There are 2 blue marbles out of 3 + 2 + 5 = 10 total marbles.
   Full step-by-step solution

   Step 1: Find the probability of drawing a blue marble. There are 2 blue marbles out of 3 + 2 + 5 = 10 total marbles. Probability of blue marble = 2/10 = 1/5 Step 2: Find the probability of rolling a number greater than 4 on a six-sided die. Numbers greater than 4 are 5 and 6. Probability of number > 4 = 2/6 = 1/3 Step 3: Since these are independent events, multiply the probabilities. Probability of both events = (1/5) × (1/3) = 1/15 The answer is 1/15.

3. P(red) = 3/8, P(blue) = 1/4, P(red or blue) = ? Answer: 5/8 Solution: Identify the given probabilities: P(red) = 3/8, P(blue) = 1/4 Convert 1/4 to eighths to match denominators: 1/4 = 2/8 Since these are mutually exclusive events (cannot both happen at the same time), add the probabilities: 3/8 + 2/8 = 5/8 The probability of drawing red or blue is 5/8 The answer…
   Full step-by-step solution

   Step 1: Identify the given probabilities: P(red) = 3/8, P(blue) = 1/4 Step 2: Convert 1/4 to eighths to match denominators: 1/4 = 2/8 Step 3: Since these are mutually exclusive events (cannot both happen at the same time), add the probabilities: 3/8 + 2/8 = 5/8 Step 4: The probability of drawing red or blue is 5/8 The answer is 5/8.

4. P(red) = 3/8 and P(blue) = 1/4, P(red or blue) = ? Answer: 5/8 Solution: P(red) = 3/8 P(blue) = 1/4 We want P(red or blue). Check if the events are mutually exclusive. The problem does not mention that red and blue can occur together, so we assume they are mutually exclusive (no overlap).
   Full step-by-step solution

   We are given: P(red) = 3/8 P(blue) = 1/4 We want P(red or blue). Step 1: Check if the events are mutually exclusive. The problem does not mention that red and blue can occur together, so we assume they are mutually exclusive (no overlap). That means: P(red or blue) = P(red) + P(blue) Step 2: Write the probabilities with a common denominator. P(red) = 3/8 P(blue) = 1/4 = 2/8 Step 3: Add the probabilities. P(red or blue) = 3/8 + 2/8 = (3 + 2)/8 = 5/8 Step 4: Conclusion. The probability of drawing red or blue is 5/8. Final answer: 5/8

5. A school is planning a field trip and needs to arrange transportation. There are 240 students and 12 teachers going on the trip. Each bus can hold 36 people. How many buses are needed for the field trip? Answer: 7 Solution: Find the total number of people going on the trip. We have 240 students and 12 teachers. Total people = 240 + 12 = 252.
   Full step-by-step solution

   Step 1: Find the total number of people going on the trip. We have 240 students and 12 teachers. Total people = 240 + 12 = 252. Step 2: Determine how many people each bus can hold. Each bus holds 36 people. Step 3: Divide the total number of people by the bus capacity to find how many buses are needed. Buses needed = 252 / 36. Step 4: Perform the division. 36 × 7 = 252, so 252 / 36 = 7 exactly. Step 5: Interpret the result. Since the division gives exactly 7, we need 7 buses to hold all 252 people. Final answer: 7 buses.

6. Emma is conducting a probability experiment with a bag of marbles and a spinner. The bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. The spinner has 4 equal sections labeled 1, 2, 3, and 4. If Emma randomly draws one marble from the bag and spins the spinner once, what is the probability that she draws a blue marble AND the spinner lands on an even number? Answer: 1/10 Solution: Find the probability of drawing a blue marble. There are 2 blue marbles out of 3 + 2 + 5 = 10 total marbles. Probability(blue marble) = 2/10 = 1/5.
   Full step-by-step solution

   Step 1: Find the probability of drawing a blue marble. There are 2 blue marbles out of 3 + 2 + 5 = 10 total marbles. Probability(blue marble) = 2/10 = 1/5. Step 2: Find the probability of the spinner landing on an even number. The even numbers on the spinner are 2 and 4. There are 2 favorable outcomes out of 4 total sections. Probability(even number) = 2/4 = 1/2. Step 3: Find the probability of both events happening. Since the events are independent, multiply the individual probabilities. Probability(blue AND even) = (1/5) × (1/2) = 1/10. The answer is 1/10.

7. (-3)² + 2 × (15 - 7) = ? Answer: 25 Solution: Handle parentheses first. Inside the parentheses: 15 - 7 = 8 (-3)² + 2 × (8) Evaluate exponents. (-3)² means (-3) × (-3) = 9 9 + 2 × 8 Perform multiplication.
   Full step-by-step solution

   Let's solve step-by-step using the order of operations (PEMDAS/BODMAS). Step 1: Handle parentheses first. Inside the parentheses: 15 - 7 = 8 So the expression becomes: (-3)² + 2 × (8) Step 2: Evaluate exponents. (-3)² means (-3) × (-3) = 9 So now we have: 9 + 2 × 8 Step 3: Perform multiplication. 2 × 8 = 16 So now we have: 9 + 16 Step 4: Perform addition. 9 + 16 = 25 Final answer: 25

8. Liam is conducting a probability experiment with a spinner divided into 4 equal sections (red, blue, green, yellow) and a standard six-sided die numbered 1-6. He wants to find the probability of spinning red on the spinner AND rolling an even number on the die. What is the probability of this compound event occurring? Answer: 1/8 Solution: Find the probability of spinning red on the spinner. The spinner has 4 equal sections: red, blue, green, yellow. Only 1 section is red.
   Full step-by-step solution

   Step 1: Find the probability of spinning red on the spinner. The spinner has 4 equal sections: red, blue, green, yellow. Only 1 section is red. So, the probability of spinning red is: P(red) = number of red sections / total number of sections = 1/4 Step 2: Find the probability of rolling an even number on the die. A standard six-sided die has faces numbered 1, 2, 3, 4, 5, 6. The even numbers are 2, 4, 6. So, there are 3 even numbers out of 6 total numbers. The probability of rolling an even number is: P(even) = number of even numbers / total number of outcomes = 3/6 = 1/2 Step 3: Find the probability of both events happening together. The problem asks for the probability of spinning red AND rolling an even number. Since these are independent events (the spinner result does not affect the die roll and vice versa), we multiply their individual probabilities. P(red and even) = P(red) * P(even) = (1/4) * (1/2) Step 4: Perform the multiplication. (1/4) * (1/2) = (1 * 1) / (4 * 2) = 1/8 Step 5: State the final answer. The probability of spinning red and rolling an even number is 1/8.