To find the probability of the intersection of two independent events [tex]\(Q\)[/tex] and [tex]\(R\)[/tex] (i.e., [tex]\( P(Q \text{ and } R) \)[/tex]), we use the rule that states:
[tex]\[ P(Q \text{ and } R) = P(Q) \times P(R) \][/tex]
Given:
- [tex]\( P(Q) = 0.78 \)[/tex]
- [tex]\( P(R) = 0.31 \)[/tex]
Since [tex]\( Q \)[/tex] and [tex]\( R \)[/tex] are independent, their joint probability is the product of their individual probabilities.
Let's calculate it step-by-step:
1. Identify the probability of [tex]\( Q \)[/tex]:
[tex]\[ P(Q) = 0.78 \][/tex]
2. Identify the probability of [tex]\( R \)[/tex]:
[tex]\[ P(R) = 0.31 \][/tex]
3. Multiply these probabilities to find [tex]\( P(Q \text{ and } R) \)[/tex]:
[tex]\[ P(Q \text{ and } R) = P(Q) \times P(R) \][/tex]
[tex]\[ P(Q \text{ and } R) = 0.78 \times 0.31 \][/tex]
Performing the multiplication yields:
[tex]\[ P(Q \text{ and } R) = 0.2418 \][/tex]
Therefore, the probability of both events [tex]\( Q \)[/tex] and [tex]\( R \)[/tex] occurring is [tex]\( 0.2418 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{0.2418} \][/tex]
