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Two friends argue over who brushes their teeth more often. To settle the argument, they keep track of the number of mornings and nights they brush and calculate a probability. These are shown in the table below:

\begin{tabular}{|c|c|c|}
\hline & Braxton & Arabella \\
\hline \begin{tabular}{c}
Probability of brushing \\
in morning
\end{tabular} & 0.79 & 0.85 \\
\hline \begin{tabular}{c}
Probability of brushing \\
in evening
\end{tabular} & 0.81 & 0.72 \\
\hline
\end{tabular}

Who is more likely to brush both morning and evening? Assume all events are independent.

A. Arabella. She has a 0.72 probability of brushing both times.
B. Braxton. He has a 0.64 probability of brushing both times.
C. Arabella. She has a 0.61 probability of brushing both times.
D. Braxton. He has a 0.79 probability of brushing both times.



Answer :

GinnyAnswer
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.

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