Compound Probability

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Compound Probability · Grade 7 · Worksheet 2

1. Emma is designing a board game that uses two special dice. The first die is a 12-sided die numbered from 1 to 12. The second die is an 8-sided die numbered from 1 to 8. If a player rolls both dice, what is the probability that they roll a number greater than 9 on the first die AND an odd number on the second die? Express your answer as a simplified fraction. Answer: 1/16 Solution: Find the probability of rolling greater than 9 on the 12-sided die.
   Full step-by-step solution

   Step 1: Find the probability of rolling greater than 9 on the 12-sided die. Numbers greater than 9 are: 10, 11, 12 (3 numbers) Total possible outcomes on 12-sided die: 12 Probability = 3/12 = 1/4 Step 2: Find the probability of rolling an odd number on the 8-sided die. Odd numbers from 1 to 8 are: 1, 3, 5, 7 (4 numbers) Total possible outcomes on 8-sided die: 8 Probability = 4/8 = 1/2 Step 3: For compound events with AND, multiply the probabilities. Probability = (1/4) × (1/2) = 1/8 Step 4: Verify the total number of possible outcomes. Total outcomes = 12 × 8 = 96 Favorable outcomes = (3 numbers > 9) × (4 odd numbers) = 3 × 4 = 12 Probability = 12/96 = 1/8 The answer is 1/8.

2. Emma is conducting a probability experiment with a bag containing 3 red marbles, 2 blue marbles, and 5 green marbles. She randomly selects one marble, records its color, and puts it back in the bag. Then she randomly selects a second marble. What is the probability that both marbles she selects are green? Express your answer as a simplified fraction. Answer: 1/4 Solution: Find the total number of marbles in the bag. 3 red + 2 blue + 5 green = 10 marbles Find the probability of selecting a green marble on the first draw.
   Full step-by-step solution

   Step 1: Find the total number of marbles in the bag. 3 red + 2 blue + 5 green = 10 marbles Step 2: Find the probability of selecting a green marble on the first draw. Number of green marbles = 5 Total marbles = 10 Probability(first green) = 5/10 = 1/2 Step 3: Since Emma puts the marble back, the bag composition remains the same for the second draw. Probability(second green) = 5/10 = 1/2 Step 4: For compound events where both events must happen, multiply the probabilities. Probability(both green) = (1/2) × (1/2) = 1/4 The answer is 1/4.

3. 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.

4. Charlotte is playing a board game where she spins a spinner with 3 equal sections labeled 1, 2, and 3, and then rolls a 6-sided die with sides labeled 1, 2, 3, 4, 5, and 6. Draw a tree diagram in your mind to represent all possible outcomes of the compound event (spin the spinner, then roll the die). How many total outcomes are in the sample space? Answer: 18 Solution: The spinner has 3 sections, so there are 3 possible outcomes for the first event. The die has 6 sides, so there are 6 possible outcomes for the second event.
   Full step-by-step solution

   Step 1: The spinner has 3 sections, so there are 3 possible outcomes for the first event. Step 2: The die has 6 sides, so there are 6 possible outcomes for the second event. Step 3: To find the total number of outcomes in the sample space for the compound event, multiply the number of outcomes for each event: 3 * 6 = 18. Step 4: The tree diagram would show 3 main branches, each splitting into 6 smaller branches, giving 18 total endpoints. The answer is 18.

5. Mere is designing a probability game using two spinners. The first spinner is divided into 4 equal sections labeled 2, 4, 6, and 8. The second spinner is divided into 3 equal sections labeled 10, 12, and 14. Draw a tree diagram in your mind to show all possible outcomes when both spinners are spun. Then, find the probability (as a simplified fraction) that the sum of the two numbers is an even number. Answer: 1/1 Solution: Identify the outcomes for each spinner. First spinner: 2, 4, 6, 8 (all even). Second spinner: 10, 12, 14 (all even).
   Full step-by-step solution

   Step 1: Identify the outcomes for each spinner. First spinner: 2, 4, 6, 8 (all even). Second spinner: 10, 12, 14 (all even). Step 2: Create the sample space using a tree diagram. For each first spinner outcome, there are 3 second spinner outcomes. Total outcomes = 4 * 3 = 12. List all pairs: (2,10), (2,12), (2,14), (4,10), (4,12), (4,14), (6,10), (6,12), (6,14), (8,10), (8,12), (8,14). Step 3: Calculate each sum: 2+10=12, 2+12=14, 2+14=16, 4+10=14, 4+12=16, 4+14=18, 6+10=16, 6+12=18, 6+14=20, 8+10=18, 8+12=20, 8+14=22. Step 4: All sums are even numbers because even + even = even. So all 12 outcomes result in an even sum. Step 5: Probability = number of favorable outcomes / total outcomes = 12/12 = 1/1. The probability is 1 (certain event).

6. 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.

7. (-4)³ ÷ 8 + 5 × (12 - 7) = ? Answer: 17 Solution: Calculate inside the parentheses: 12 - 7 = 5 Calculate the exponent: (-4)³ = -4 × -4 × -4 = 16 × -4 = -64 Perform division: -64 ÷ 8 = -8 Perform multiplication: 5 × 5 = 25 Add the results: -8 + 25 = 17 The answer is 17.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: 12 - 7 = 5 Step 2: Calculate the exponent: (-4)³ = -4 × -4 × -4 = 16 × -4 = -64 Step 3: Perform division: -64 ÷ 8 = -8 Step 4: Perform multiplication: 5 × 5 = 25 Step 5: Add the results: -8 + 25 = 17 The answer is 17.