To determine who is more likely to brush their teeth both in the morning and in the evening, we need to find the joint probability for each person. This can be calculated by multiplying the individual probabilities of brushing in the morning and brushing in the evening, assuming these events are independent.
### Step-by-Step Solution:
1. Given Probabilities:
- For Braxton:
- Probability of brushing in the morning (P(M_braxton)) = 0.72
- Probability of brushing in the evening (P(E_braxton)) = 0.85
- For Arabella:
- Probability of brushing in the morning (P(M_arabella)) = 0.82
- Probability of brushing in the evening (P(E_arabella)) = 0.79
2. Joint Probability Calculation:
- The probability of both events occurring (brushing both morning and evening) is found by multiplying the individual probabilities.
- For Braxton:
[tex]\[
P(\text{Both\_Braxton}) = P(M_{\text{braxton}}) \times P(E_{\text{braxton}}) = 0.72 \times 0.85
\][/tex]
Substituting the values:
[tex]\[
P(\text{Both\_Braxton}) = 0.72 \times 0.85 = 0.612
\][/tex]
- For Arabella:
[tex]\[
P(\text{Both\_Arabella}) = P(M_{\text{arabella}}) \times P(E_{\text{arabella}}) = 0.82 \times 0.79
\][/tex]
Substituting the values:
[tex]\[
P(\text{Both\_Arabella}) = 0.82 \times 0.79 = 0.6478
\][/tex]
3. Comparison:
To determine who is more likely to brush both morning and evening, we compare the joint probabilities.
- [tex]\[
P(\text{Both\_Braxton}) = 0.612
\][/tex]
- [tex]\[
P(\text{Both\_Arabella}) = 0.6478
\][/tex]
Since [tex]\(0.6478 > 0.612\)[/tex], Arabella has a higher probability of brushing both morning and evening.
### Conclusion:
Arabella is more likely to brush her teeth both in the morning and in the evening.
