Compound Probability

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Compound Probability · Grade 7 · Worksheet 3

1. 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.

2. P(red) = 3/8 and P(blue) = 1/6, P(red or blue) = ? Answer: 13/24 Solution: Write down the given probabilities: P(red) = 3/8, P(blue) = 1/6 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) Substitute the values: P(red or blue) = 3/8 + 1/6 Find a common denominator for the fractions.
   Full step-by-step solution

   Step 1: Write down the given probabilities: P(red) = 3/8, P(blue) = 1/6 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: Substitute the values: P(red or blue) = 3/8 + 1/6 Step 4: Find a common denominator for the fractions. The least common multiple of 8 and 6 is 24. Step 5: Convert 3/8 to have denominator 24: 3/8 = (3 × 3)/(8 × 3) = 9/24 Step 6: Convert 1/6 to have denominator 24: 1/6 = (1 × 4)/(6 × 4) = 4/24 Step 7: Add the fractions: 9/24 + 4/24 = 13/24 Step 8: The probability cannot be simplified further, so the final answer is 13/24.

3. Emma is conducting a probability experiment with a bag containing 5 red marbles, 3 blue marbles, and 2 green marbles. She draws one marble at random, records its color, and puts it back in the bag. Then she draws a second marble. What is the probability that Emma draws a red marble first AND a blue marble second? Express your answer as a simplified fraction. Answer: 3/20 Solution: Calculate the total number of marbles. Total marbles = 5 red + 3 blue + 2 green = 10 marbles. Calculate the probability of drawing a red marble first.
   Full step-by-step solution

   Step 1: Calculate the total number of marbles. Total marbles = 5 red + 3 blue + 2 green = 10 marbles. Step 2: Calculate the probability of drawing a red marble first. P(red first) = number of red marbles / total marbles = 5/10 = 1/2. Step 3: Since Emma puts the marble back, the total number of marbles remains the same for the second draw. Calculate the probability of drawing a blue marble second. P(blue second) = number of blue marbles / total marbles = 3/10. Step 4: Calculate the probability of both events happening (red first AND blue second). P(red first AND blue second) = P(red first) × P(blue second) = (1/2) × (3/10) = 3/20. The answer is 3/20.

4. P(red or blue) = 3/10 + 1/5 = ? Answer: 1/2 Solution: P(red or blue) = 3/10 + 1/5 This means we are adding two probabilities. The first fraction is 3/10. The second fraction is 1/5.
   Full step-by-step solution

   Step 1: Understand the problem We are given: P(red or blue) = 3/10 + 1/5 This means we are adding two probabilities. Step 2: Check if the fractions have the same denominator The first fraction is 3/10. The second fraction is 1/5. Since the denominators are different (10 and 5), we cannot add them directly. Step 3: Find a common denominator The least common denominator of 10 and 5 is 10. We convert 1/5 to a fraction with denominator 10: 1/5 = (1 × 2)/(5 × 2) = 2/10. Step 4: Add the fractions Now we have: 3/10 + 2/10 = (3 + 2)/10 = 5/10. Step 5: Simplify the result 5/10 can be simplified by dividing numerator and denominator by 5: 5 ÷ 5 = 1, 10 ÷ 5 = 2, so 5/10 = 1/2. Step 6: Final answer P(red or blue) = 1/2.

5. (-4)² - 3 × (18 ÷ 3 - 2) = ? Answer: 4 Solution: Evaluate inside the parentheses: 18 ÷ 3 - 2 = 6 - 2 = 4 Apply the exponent: (-4)² = 16 Perform the multiplication: 3 × 4 = 12 Subtract: 16 - 12 = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Evaluate inside the parentheses: 18 ÷ 3 - 2 = 6 - 2 = 4 Step 2: Apply the exponent: (-4)² = 16 Step 3: Perform the multiplication: 3 × 4 = 12 Step 4: Subtract: 16 - 12 = 4 The answer is 4.

6. Liam is conducting a probability experiment with two spinners. The first spinner has 4 equal sections labeled A, B, C, and D. The second spinner has 3 equal sections labeled Red, Blue, and Green. If Liam spins both spinners once, what is the probability that he lands on section B and the color Red? Answer: 1/12 Solution: Determine the total number of possible outcomes. The first spinner has 4 equally likely outcomes: A, B, C, D. The second spinner has 3 equally likely outcomes: Red, Blue, Green.
   Full step-by-step solution

   Let's go step-by-step. Step 1: Determine the total number of possible outcomes. The first spinner has 4 equally likely outcomes: A, B, C, D. The second spinner has 3 equally likely outcomes: Red, Blue, Green. When spinning both spinners, the total number of possible outcomes is: 4 × 3 = 12. Step 2: Identify the favorable outcome. We want the outcome where the first spinner lands on B and the second spinner lands on Red. That is only 1 specific outcome: (B, Red). Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of possible outcomes) Probability = 1 / 12. Step 4: Conclusion. The probability that Liam lands on B and Red is 1/12.

7. (-3)² - 4 × (2 - 5) = ? Answer: 21 Solution: Expression: (-3)² - 4 × (2 - 5) Handle the exponent first. (-3)² means (-3) × (-3) = 9 So now we have: 9 - 4 × (2 - 5) Evaluate inside the parentheses.
   Full step-by-step solution

   Let's solve step-by-step: Expression: (-3)² - 4 × (2 - 5) Step 1: Handle the exponent first. (-3)² means (-3) × (-3) = 9 So now we have: 9 - 4 × (2 - 5) Step 2: Evaluate inside the parentheses. (2 - 5) = -3 Now we have: 9 - 4 × (-3) Step 3: Perform multiplication before subtraction (order of operations: PEMDAS/BODMAS). 4 × (-3) = -12 Now we have: 9 - (-12) Step 4: Subtracting a negative is the same as adding a positive. 9 - (-12) = 9 + 12 = 21 Final Answer: 21

8. Emma is conducting a probability experiment with a bag containing 4 red marbles, 3 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 fraction in simplest form. Answer: 25/144 Solution: Find the total number of marbles. There are 4 red + 3 blue + 5 green = 12 marbles total. 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. There are 4 red + 3 blue + 5 green = 12 marbles total. Step 2: Find the probability of selecting a green marble on the first draw. There are 5 green marbles out of 12 total, so the probability is 5/12. Step 3: Since Emma puts the marble back, the bag has the same composition for the second draw. The probability of selecting a green marble on the second draw is also 5/12. Step 4: To find the probability of both events happening (green AND green), multiply the probabilities: (5/12) × (5/12) = 25/144. Step 5: Check if the fraction can be simplified. The greatest common factor of 25 and 144 is 1, so 25/144 is already in simplest form. The answer is 25/144.