First, let's identify the probability of each individual event:
1. Event A: The first coin lands on heads.
- Since a coin has two sides, the probability of landing on heads is [tex]\( \frac{1}{2} \)[/tex].
2. Event B: The second coin lands on tails.
- Similar to the first coin, the probability of landing on tails is also [tex]\( \frac{1}{2} \)[/tex].
Next, because these two events (the outcomes of two coin tosses) are independent, the probability of both events occurring together (both coins showing the specified outcomes) is the product of their individual probabilities. This can be calculated using the formula for independent events:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Substituting the given probabilities:
[tex]\[ P(A) = \frac{1}{2} \][/tex]
[tex]\[ P(B) = \frac{1}{2} \][/tex]
Thus,
[tex]\[ P(A \text{ and } B) = \frac{1}{2} \cdot \frac{1}{2} \][/tex]
Performing the multiplication:
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
Therefore, the probability that both events will occur (the first coin lands on heads and the second coin lands on tails) is:
[tex]\[ P(A \text{ and } B) = \frac{1}{4} \][/tex]
