Answer :
To solve this problem, let's break it down step by step:
1. Event A: The coin lands on heads.
The probability of a coin landing on heads (event A) can be calculated since a fair coin has two possible outcomes: heads or tails. Therefore, the probability of the coin landing on heads is:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
2. Event B: The die lands on 1, 3, or 6.
A standard six-sided die has six faces, each numbered from 1 to 6. To find the probability of the die landing on 1, 3, or 6, note that these are three of the six possible outcomes.
[tex]\[ P(B) = \frac{3}{6} = \frac{1}{2} \][/tex]
3. Both Events Occurring: The product of their probabilities.
Since tossing a coin and rolling a die are independent events, the probability that both events A and B will occur can be calculated by multiplying the probability of each event:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
4. Substitute the probabilities into the formula:
[tex]\[ P(A \text{ and } B) = \left( \frac{1}{2} \right) \cdot \left( \frac{1}{2} \right) \][/tex]
Multiplying these fractions:
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4} \][/tex]
So, the probability that both events will occur is:
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
1. Event A: The coin lands on heads.
The probability of a coin landing on heads (event A) can be calculated since a fair coin has two possible outcomes: heads or tails. Therefore, the probability of the coin landing on heads is:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
2. Event B: The die lands on 1, 3, or 6.
A standard six-sided die has six faces, each numbered from 1 to 6. To find the probability of the die landing on 1, 3, or 6, note that these are three of the six possible outcomes.
[tex]\[ P(B) = \frac{3}{6} = \frac{1}{2} \][/tex]
3. Both Events Occurring: The product of their probabilities.
Since tossing a coin and rolling a die are independent events, the probability that both events A and B will occur can be calculated by multiplying the probability of each event:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
4. Substitute the probabilities into the formula:
[tex]\[ P(A \text{ and } B) = \left( \frac{1}{2} \right) \cdot \left( \frac{1}{2} \right) \][/tex]
Multiplying these fractions:
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4} \][/tex]
So, the probability that both events will occur is:
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
