Certainly! Let's solve the problem step-by-step:
1. Identify the given probabilities:
- The probability of event \( P \) occurring, denoted as \( P(P) \), is given as \( \frac{2}{3} \).
- The probability of event \( Q \) occurring, denoted as \( P(Q) \), is given as \( \frac{3}{4} \).
2. Assume the events are independent:
When two events are independent, the probability of both events occurring together, denoted as \( P(P \cap Q) \), is the product of their individual probabilities.
3. Calculate the joint probability:
- To find \( P(P \cap Q) \), multiply the probabilities \( P(P) \) and \( P(Q) \).
[tex]\[
P(P \cap Q) = P(P) \times P(Q)
\][/tex]
[tex]\[
P(P \cap Q) = \left(\frac{2}{3}\right) \times \left(\frac{3}{4}\right)
\][/tex]
4. Simplify the product:
- Multiply the numerators together and the denominators together:
[tex]\[
P(P \cap Q) = \frac{2 \times 3}{3 \times 4}
\][/tex]
[tex]\[
P(P \cap Q) = \frac{6}{12}
\][/tex]
- Simplify the fraction \( \frac{6}{12} \):
[tex]\[
\frac{6}{12} = \frac{1}{2}
\][/tex]
So the probability of both events \( P \) and \( Q \) occurring is \( \frac{1}{2} \) or 0.5.
Thus, the probability of the outcomes [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] is [tex]\( 0.5 \)[/tex].
