Answer :
To solve this problem, let's go through the details step-by-step:
1. Determine the probability of Event A (Flipping heads on a coin):
A fair coin has two sides: heads and tails. The probability of flipping heads, which we'll call \( P(A) \), is given by the ratio of favorable outcomes to the total possible outcomes. Since there is only one favorable outcome (heads) out of two possible outcomes (heads or tails), the probability is:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
2. Determine the probability of Event B (Rolling an odd number on a number cube):
A fair number cube (or a standard 6-sided die) has six faces, numbered 1 through 6. The odd numbers on this cube are 1, 3, and 5. There are three favorable outcomes (rolling a 1, 3, or 5) out of six possible outcomes. Therefore, the probability of rolling an odd number, which we'll call \( P(B) \), is:
[tex]\[ P(B) = \frac{3}{6} = \frac{1}{2} \][/tex]
3. Calculate the joint probability of both events happening (Event A and Event B):
Since flipping a coin and rolling a number cube are independent events, the probability of both events occurring together (i.e., flipping heads and rolling an odd number) can be found by multiplying their individual probabilities. This joint probability is represented as \( P(A \text{ and } B) \):
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
Substituting the probabilities, we get:
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \][/tex]
Therefore, the probability of flipping heads on a coin and rolling an odd number on a number cube is:
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
The correct answer is [tex]\( \boxed{\frac{1}{4}} \)[/tex], which corresponds to option C in the given multiple-choice options.
1. Determine the probability of Event A (Flipping heads on a coin):
A fair coin has two sides: heads and tails. The probability of flipping heads, which we'll call \( P(A) \), is given by the ratio of favorable outcomes to the total possible outcomes. Since there is only one favorable outcome (heads) out of two possible outcomes (heads or tails), the probability is:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
2. Determine the probability of Event B (Rolling an odd number on a number cube):
A fair number cube (or a standard 6-sided die) has six faces, numbered 1 through 6. The odd numbers on this cube are 1, 3, and 5. There are three favorable outcomes (rolling a 1, 3, or 5) out of six possible outcomes. Therefore, the probability of rolling an odd number, which we'll call \( P(B) \), is:
[tex]\[ P(B) = \frac{3}{6} = \frac{1}{2} \][/tex]
3. Calculate the joint probability of both events happening (Event A and Event B):
Since flipping a coin and rolling a number cube are independent events, the probability of both events occurring together (i.e., flipping heads and rolling an odd number) can be found by multiplying their individual probabilities. This joint probability is represented as \( P(A \text{ and } B) \):
[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]
Substituting the probabilities, we get:
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \][/tex]
Therefore, the probability of flipping heads on a coin and rolling an odd number on a number cube is:
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
The correct answer is [tex]\( \boxed{\frac{1}{4}} \)[/tex], which corresponds to option C in the given multiple-choice options.
