Answer :
Sure, let's solve this step by step.
### Step 1: Understand the Probability of Event A
Event A is that the coin lands on heads. Since a fair coin has two sides (heads and tails), the probability of the coin landing on heads is:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
### Step 2: Understand the Probability of Event B
Event B is that the die lands on 1, 3, or 6. A fair six-sided die has six faces, numbered 1 through 6. The favorable outcomes for Event B are:
- Landing on 1
- Landing on 3
- Landing on 6
So, there are 3 favorable outcomes out of a total of 6 possible outcomes. Therefore, the probability of Event B is:
[tex]\[ P(B) = \frac{3}{6} \][/tex]
This fraction simplifies to:
[tex]\[ P(B) = \frac{1}{2} \][/tex]
### Step 3: Combining the Probabilities of Independent Events
To find the probability that both events will occur (i.e., the coin lands on heads and the die lands on 1, 3, or 6), we use the formula for the probability of independent events:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
### Step 4: Calculate the Combined Probability
Now, substitute the probabilities we found:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
[tex]\[ P(B) = \frac{1}{2} \][/tex]
Therefore:
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{2} \][/tex]
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
The probability that both the coin will land on heads and the die will land on 1, 3, or 6 is:
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
In decimal form, this is:
[tex]\[ P(A \text{ and } B) = 0.25 \][/tex]
So, the answer in its simplest form is:
[tex]\[ \boxed{\frac{1}{4}} \][/tex]
### Step 1: Understand the Probability of Event A
Event A is that the coin lands on heads. Since a fair coin has two sides (heads and tails), the probability of the coin landing on heads is:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
### Step 2: Understand the Probability of Event B
Event B is that the die lands on 1, 3, or 6. A fair six-sided die has six faces, numbered 1 through 6. The favorable outcomes for Event B are:
- Landing on 1
- Landing on 3
- Landing on 6
So, there are 3 favorable outcomes out of a total of 6 possible outcomes. Therefore, the probability of Event B is:
[tex]\[ P(B) = \frac{3}{6} \][/tex]
This fraction simplifies to:
[tex]\[ P(B) = \frac{1}{2} \][/tex]
### Step 3: Combining the Probabilities of Independent Events
To find the probability that both events will occur (i.e., the coin lands on heads and the die lands on 1, 3, or 6), we use the formula for the probability of independent events:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
### Step 4: Calculate the Combined Probability
Now, substitute the probabilities we found:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
[tex]\[ P(B) = \frac{1}{2} \][/tex]
Therefore:
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{2} \][/tex]
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
The probability that both the coin will land on heads and the die will land on 1, 3, or 6 is:
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
In decimal form, this is:
[tex]\[ P(A \text{ and } B) = 0.25 \][/tex]
So, the answer in its simplest form is:
[tex]\[ \boxed{\frac{1}{4}} \][/tex]
