To determine who is more likely to brush both morning and evening, we need to calculate the joint probability of brushing in both the morning and evening for both Braxton and Arabella. We assume that the events of brushing in the morning and in the evening are independent.
### Step-by-Step Solution:
#### 1. Calculate the joint probability for Braxton:
- Probability of brushing in the morning (Braxton): [tex]\( P(\text{Morning, Braxton}) = 0.79 \)[/tex]
- Probability of brushing in the evening (Braxton): [tex]\( P(\text{Evening, Braxton}) = 0.81 \)[/tex]
Since these events are independent, we can find the joint probability by multiplying these probabilities:
[tex]\[ P(\text{Both, Braxton}) = P(\text{Morning, Braxton}) \times P(\text{Evening, Braxton}) \][/tex]
[tex]\[ P(\text{Both, Braxton}) = 0.79 \times 0.81 = 0.6399 \][/tex]
#### 2. Calculate the joint probability for Arabella:
- Probability of brushing in the morning (Arabella): [tex]\( P(\text{Morning, Arabella}) = 0.85 \)[/tex]
- Probability of brushing in the evening (Arabella): [tex]\( P(\text{Evening, Arabella}) = 0.72 \)[/tex]
Similarly, we find the joint probability for Arabella:
[tex]\[ P(\text{Both, Arabella}) = P(\text{Morning, Arabella}) \times P(\text{Evening, Arabella}) \][/tex]
[tex]\[ P(\text{Both, Arabella}) = 0.85 \times 0.72 = 0.612 \][/tex]
#### 3. Compare the probabilities:
- Braxton's joint probability: [tex]\( 0.6399 \)[/tex]
- Arabella's joint probability: [tex]\( 0.612 \)[/tex]
Braxton has a higher probability of brushing both morning and evening compared to Arabella.
### Conclusion:
Braxton is more likely to brush both morning and evening with a probability of [tex]\( 0.6399 \)[/tex].
The correct answer is:
B. Braxton. He has a 0.64 probability of brushing both times.
