P(red) = 3/8 and P(blue) = 1/6, P(red or blue) = ?Answer: ______________
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: ______________
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: ______________
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: ______________
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
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.
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.
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.
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.
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.
(-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
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