To find the probability that both events A and B will occur, let's break down the problem step by step.
### Step 1: Determine the probability of Event A
Event A is that the coin lands on tails.
- A standard coin has two sides: heads and tails.
- The probability of the coin landing on tails, [tex]\( P(A) \)[/tex], is:
[tex]\[
P(A) = \frac{1 \text{ favorable outcome}}{2 \text{ possible outcomes}} = \frac{1}{2} = 0.5
\][/tex]
### Step 2: Determine the probability of Event B
Event B is that the die lands on either 4 or 5.
- A standard six-sided die has six faces numbered from 1 to 6.
- There are two favorable outcomes for Event B: landing on 4 or 5.
- Therefore, the probability of the die landing on 4 or 5, [tex]\( P(B) \)[/tex], is:
[tex]\[
P(B) = \frac{2 \text{ favorable outcomes}}{6 \text{ possible outcomes}} = \frac{2}{6} = \frac{1}{3} \approx 0.3333
\][/tex]
### Step 3: Determine the probability of both events occurring
Since the events of the coin landing on tails and the die landing on 4 or 5 are independent, we can use the multiplication rule for independent events.
- The probability of both events, [tex]\( P(A \text{ and } B) \)[/tex], is given by:
[tex]\[
P(A \text{ and } B) = P(A) \cdot P(B)
\][/tex]
Substituting the probabilities we have found:
[tex]\[
P(A \text{ and } B) = 0.5 \times \frac{1}{3} = \frac{0.5 \cdot 1}{1 \cdot 3} = \frac{1}{6} \approx 0.1667
\][/tex]
Thus, the probability that both the coin lands on tails and the die lands on 4 or 5 is:
[tex]\[
P(A \text{ and } B) = \frac{1}{6}
\][/tex]
This simplifies the answer in its simplest form.
[tex]\[
\boxed{\frac{1}{6}}
\][/tex]
