To determine who is more likely to brush both morning and evening, we need to calculate the combined probability for each person by multiplying the individual probabilities of brushing in the morning and evening, given that the events are independent.
### Step-by-Step Solution:
1. Calculate the combined probability for Braxton:
- Probability of brushing in the morning (Braxton): [tex]\( P_{\text{morning, Braxton}} = 0.72 \)[/tex]
- Probability of brushing in the evening (Braxton): [tex]\( P_{\text{evening, Braxton}} = 0.85 \)[/tex]
- Combined probability for Braxton [tex]\( P_{\text{both, Braxton}} \)[/tex]:
[tex]\[
P_{\text{both, Braxton}} = P_{\text{morning, Braxton}} \times P_{\text{evening, Braxton}} = 0.72 \times 0.85 = 0.612
\][/tex]
2. Calculate the combined probability for Arabella:
- Probability of brushing in the morning (Arabella): [tex]\( P_{\text{morning, Arabella}} = 0.82 \)[/tex]
- Probability of brushing in the evening (Arabella): [tex]\( P_{\text{evening, Arabella}} = 0.79 \)[/tex]
- Combined probability for Arabella [tex]\( P_{\text{both, Arabella}} \)[/tex]:
[tex]\[
P_{\text{both, Arabella}} = P_{\text{morning, Arabella}} \times P_{\text{evening, Arabella}} = 0.82 \times 0.79 = 0.6478
\][/tex]
3. Comparison of probabilities:
- Braxton's combined probability of brushing both times is [tex]\( 0.612 \)[/tex].
- Arabella's combined probability of brushing both times is [tex]\( 0.6478 \)[/tex].
4. Determine the more likely scenario:
- Since 0.6478 (Arabella) is greater than 0.612 (Braxton), Arabella is more likely to brush both in the morning and evening.
Given the calculations, Arabella has a higher combined probability of brushing both morning and evening.
Conclusion:
The correct answer is:
B. Arabella. She has a 0.65 probability of brushing both times.
