Compound Probability

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Answer Key & Explanations

Compound Probability · Grade 7 · Worksheet 1

1. Liam is conducting a probability experiment with a standard six-sided die and a fair coin. He wants to determine the probability of rolling an even number on the die and getting heads on the coin flip. What is the probability of this compound event occurring? Answer: 1/4 Solution: Identify the two independent events. Event A: Rolling an even number on a standard six-sided die. A standard die has numbers 1, 2, 3, 4, 5, 6.
   Full step-by-step solution

   Let's solve this step by step. Step 1: Identify the two independent events. Event A: Rolling an even number on a standard six-sided die. Event B: Getting heads on a fair coin flip. Step 2: Find the probability of rolling an even number. A standard die has numbers 1, 2, 3, 4, 5, 6. Even numbers are 2, 4, 6. So, favorable outcomes = 3, total outcomes = 6. Probability(A) = 3/6 = 1/2. Step 3: Find the probability of getting heads on a coin flip. A fair coin has two equally likely outcomes: heads or tails. Favorable outcomes = 1, total outcomes = 2. Probability(B) = 1/2. Step 4: Since the die roll and coin flip are independent events, multiply their probabilities to find the compound probability. Probability(A and B) = Probability(A) × Probability(B) = (1/2) × (1/2) = 1/4. Step 5: Conclusion. The probability of rolling an even number and getting heads is 1/4.

2. 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.

3. A circular dartboard has a radius of 15 cm. The bullseye is a smaller circle in the center with a radius of 3 cm. If a dart hits a random point on the dartboard, what is the probability (as a simplified fraction) that it lands in the bullseye? Answer: 1/25 Solution: We have a circular dartboard with radius 15 cm. The bullseye is a smaller circle in the center with radius 3 cm. A dart hits a random point on the dartboard.
   Full step-by-step solution

   Step 1: Understand the problem. We have a circular dartboard with radius 15 cm. The bullseye is a smaller circle in the center with radius 3 cm. A dart hits a random point on the dartboard. The probability of hitting the bullseye is the ratio of the area of the bullseye to the area of the whole dartboard. Step 2: Recall the formula for the area of a circle. The area of a circle is given by: Area = pi * (radius)^2. Step 3: Calculate the area of the entire dartboard. Dartboard radius, R = 15 cm. Area of dartboard = pi * R^2 = pi * (15)^2 = pi * 225. Step 4: Calculate the area of the bullseye. Bullseye radius, r = 3 cm. Area of bullseye = pi * r^2 = pi * (3)^2 = pi * 9. Step 5: Set up the probability. Probability = (Area of bullseye) / (Area of dartboard) = (pi * 9) / (pi * 225). Step 6: Simplify the expression. The pi in the numerator and denominator cancel out. So, Probability = 9 / 225. Step 7: Simplify the fraction. Find the greatest common divisor (GCD) of 9 and 225. Since 9 divides 225 (because 225 / 9 = 25), the GCD is 9. Divide numerator and denominator by 9: (9 / 9) / (225 / 9) = 1 / 25. Step 8: Final answer. The probability that the dart lands in the bullseye is 1/25.

4. P(red) = 3/10, P(blue) = 1/4, P(red or blue) = ? Answer: 11/20 Solution: Write down the given probabilities: P(red) = 3/10, P(blue) = 1/4 Since these are mutually exclusive events (a marble cannot be both red and blue), we add the probabilities: P(red or blue) = P(red) + P(blue) Convert fractions to have a common denominator: 3/10 = 6/20, 1/4 = 5/20 Add the…
   Full step-by-step solution

   Step 1: Write down the given probabilities: P(red) = 3/10, P(blue) = 1/4 Step 2: Since these are mutually exclusive events (a marble cannot be both red and blue), we add the probabilities: P(red or blue) = P(red) + P(blue) Step 3: Convert fractions to have a common denominator: 3/10 = 6/20, 1/4 = 5/20 Step 4: Add the fractions: 6/20 + 5/20 = 11/20 Step 5: The probability of drawing either a red or blue marble is 11/20.

5. (-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.

6. Emma is conducting a probability experiment with two spinners. The first spinner has 6 equal sections labeled with the numbers 1 through 6. The second spinner has 5 equal sections labeled with the letters A, B, C, D, and E. If Emma spins both spinners once, what is the probability that she gets an even number on the first spinner AND a vowel on the second spinner? Express your answer as a simplified fraction. Answer: 1/5 Solution: Find the probability of getting an even number on the first spinner. The first spinner has numbers 1, 2, 3, 4, 5, 6. Even numbers are 2, 4, 6.
   Full step-by-step solution

   Step 1: Find the probability of getting an even number on the first spinner. The first spinner has numbers 1, 2, 3, 4, 5, 6. Even numbers are 2, 4, 6. So there are 3 favorable outcomes out of 6 total sections. Probability = 3/6 = 1/2 Step 2: Find the probability of getting a vowel on the second spinner. The second spinner has letters A, B, C, D, E. Vowels are A and E. So there are 2 favorable outcomes out of 5 total sections. Probability = 2/5 Step 3: Find the probability of both events occurring. For independent events, multiply the probabilities. Probability = (1/2) × (2/5) = 2/10 = 1/5 The answer is 1/5.