Answer :
To determine the probability that both events A and B will occur, we first need to determine the probability of each event individually and then use the rule for independent events.
### Step 1: Determine the probability of Event A.
Event A is the coin landing on heads. A fair coin has two outcomes: heads or tails. The probability of the coin landing on heads is therefore:
[tex]\[ P(A) = \frac{1}{2} = 0.5 \][/tex]
### Step 2: Determine the probability of Event B.
Event B is the die landing on either 2, 4, 5, or 6. A standard six-sided die has six outcomes: 1, 2, 3, 4, 5, and 6. There are 4 favorable outcomes (2, 4, 5, or 6) out of the 6 possible outcomes. Thus, the probability of the die landing on one of these numbers is:
[tex]\[ P(B) = \frac{4}{6} = \frac{2}{3} \approx 0.6666666666666666 \][/tex]
### Step 3: Calculate the probability of both events occurring (Event A and Event B).
For two independent events, the probability that both events occur is the product of their individual probabilities. Therefore:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Substituting the values we found:
[tex]\[ P(A \text{ and } B) = 0.5 \times \frac{2}{3} \][/tex]
Multiplying these together:
[tex]\[ P(A \text{ and } B) = 0.5 \times 0.6666666666666666 = 0.3333333333333333 = \frac{1}{3} \][/tex]
Thus, the probability that both events will occur is:
[tex]\[ \boxed{\frac{1}{3}} \][/tex]
In simplest form, the probability [tex]\(P(A \text{ and } B)\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
### Step 1: Determine the probability of Event A.
Event A is the coin landing on heads. A fair coin has two outcomes: heads or tails. The probability of the coin landing on heads is therefore:
[tex]\[ P(A) = \frac{1}{2} = 0.5 \][/tex]
### Step 2: Determine the probability of Event B.
Event B is the die landing on either 2, 4, 5, or 6. A standard six-sided die has six outcomes: 1, 2, 3, 4, 5, and 6. There are 4 favorable outcomes (2, 4, 5, or 6) out of the 6 possible outcomes. Thus, the probability of the die landing on one of these numbers is:
[tex]\[ P(B) = \frac{4}{6} = \frac{2}{3} \approx 0.6666666666666666 \][/tex]
### Step 3: Calculate the probability of both events occurring (Event A and Event B).
For two independent events, the probability that both events occur is the product of their individual probabilities. Therefore:
[tex]\[ P(A \text{ and } B) = P(A) \cdot P(B) \][/tex]
Substituting the values we found:
[tex]\[ P(A \text{ and } B) = 0.5 \times \frac{2}{3} \][/tex]
Multiplying these together:
[tex]\[ P(A \text{ and } B) = 0.5 \times 0.6666666666666666 = 0.3333333333333333 = \frac{1}{3} \][/tex]
Thus, the probability that both events will occur is:
[tex]\[ \boxed{\frac{1}{3}} \][/tex]
In simplest form, the probability [tex]\(P(A \text{ and } B)\)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
