Given the probabilities:
- [tex]\( P(R) = \frac{1}{4} \)[/tex]
- [tex]\( P(W) = \frac{8}{23} \)[/tex]
- [tex]\( P(R \text{ and } W) = \frac{2}{23} \)[/tex]
To determine whether the events [tex]\( R \)[/tex] and [tex]\( W \)[/tex] are independent, we need to check if [tex]\( P(R \text{ and } W) = P(R) \times P(W) \)[/tex].
Calculate [tex]\( P(R) \times P(W) \)[/tex]:
[tex]\[
P(R) \times P(W) = \left( \frac{1}{4} \right) \times \left( \frac{8}{23} \right) = \frac{1 \times 8}{4 \times 23} = \frac{8}{92} = \frac{2}{23}
\][/tex]
We see that:
[tex]\[
P(R \text{ and } W) = \frac{2}{23}
\][/tex]
[tex]\[
P(R) \times P(W) = \frac{2}{23}
\][/tex]
Since [tex]\( P(R \text{ and } W) = P(R) \times P(W) \)[/tex], the events [tex]\( R \)[/tex] and [tex]\( W \)[/tex] are independent.
Therefore, the completed sentence is:
The events that Elena chooses a red flower and Glenn chooses a white flower are independent because [tex]\( P(R \text{ and } W) \)[/tex] is equal to [tex]\( P(R) \ times \( P(W) \)[/tex].
