Answer :
Let's walk through the given question step-by-step to find the probability that both events (A and B) will occur.
### Step 1: Determine the Probability of Event A
Event A is defined as the coin landing on tails.
Since a fair coin has two sides (heads and tails), the probability of the coin landing on tails (Event A) can be calculated as:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
### Step 2: Determine the Probability of Event B
Event B is defined as the die landing on either a 4 or a 5.
A fair six-sided die has six faces numbered 1 through 6. The favorable outcomes for Event B are landing on 4 or 5. There are 2 favorable outcomes (4 and 5) out of the 6 possible outcomes when rolling the die. Therefore, the probability of Event B can be calculated as:
[tex]\[ P(B) = \frac{2}{6} = \frac{1}{3} \][/tex]
### Step 3: Determine the Probability of Both Events Occurring
Since the coin toss and the die roll are independent events, the probability of both events occurring together (both A and B) is the product of the individual probabilities of each event.
Using the formula for the probability of independent events:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Substituting the probabilities we found:
[tex]\[ P(A \text{ and } B) = \left( \frac{1}{2} \right) \cdot \left( \frac{1}{3} \right) \][/tex]
### Step 4: Calculate the Result
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \][/tex]
### Final Answer
The probability that both events (the coin lands on tails and the die lands on 4 or 5) will occur is:
[tex]\[ \boxed{\frac{1}{6}} \][/tex]
### Step 1: Determine the Probability of Event A
Event A is defined as the coin landing on tails.
Since a fair coin has two sides (heads and tails), the probability of the coin landing on tails (Event A) can be calculated as:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
### Step 2: Determine the Probability of Event B
Event B is defined as the die landing on either a 4 or a 5.
A fair six-sided die has six faces numbered 1 through 6. The favorable outcomes for Event B are landing on 4 or 5. There are 2 favorable outcomes (4 and 5) out of the 6 possible outcomes when rolling the die. Therefore, the probability of Event B can be calculated as:
[tex]\[ P(B) = \frac{2}{6} = \frac{1}{3} \][/tex]
### Step 3: Determine the Probability of Both Events Occurring
Since the coin toss and the die roll are independent events, the probability of both events occurring together (both A and B) is the product of the individual probabilities of each event.
Using the formula for the probability of independent events:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Substituting the probabilities we found:
[tex]\[ P(A \text{ and } B) = \left( \frac{1}{2} \right) \cdot \left( \frac{1}{3} \right) \][/tex]
### Step 4: Calculate the Result
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \][/tex]
### Final Answer
The probability that both events (the coin lands on tails and the die lands on 4 or 5) will occur is:
[tex]\[ \boxed{\frac{1}{6}} \][/tex]
