Certainly! Let's carefully work through this problem step-by-step.
First, we identify the given probabilities:
- The probability that Amber's bus arrives on time is [tex]\(\frac{5}{6}\)[/tex].
- If the bus arrives on time, the probability that Amber arrives at work by 9 am is [tex]\(\frac{4}{5}\)[/tex].
- If the bus is late, the probability that Amber arrives at work by 9 am is [tex]\(\frac{3}{10}\)[/tex].
The goal is to find the probability that Amber does not arrive at work by 9 am.
We will use the law of total probability to solve this problem.
### Step 1: Calculate the probability that Amber does not arrive by 9 am if the bus is on time.
If the bus is on time, the probability that Amber arrives by 9 am is [tex]\(\frac{4}{5}\)[/tex]. Therefore, the probability that she does not arrive by 9 am if the bus is on time is:
[tex]\[
1 - \frac{4}{5} = \frac{1}{5}
\][/tex]
### Step 2: Calculate the probability that Amber does not arrive by 9 am if the bus is late.
If the bus is late, the probability that Amber arrives by 9 am is [tex]\(\frac{3}{10}\)[/tex]. Therefore, the probability that she does not arrive by 9 am if the bus is late is:
[tex]\[
1 - \frac{3}{10} = \frac{7}{10}
\][/tex]
### Step 3: Calculate the probability that the bus is late.
The probability that the bus arrives on time is [tex]\(\frac{5}{6}\)[/tex], so the probability that the bus is late is:
[tex]\[
1 - \frac{5}{6} = \frac{1}{6}
\][/tex]
### Step 4: Combine the probabilities using the law of total probability.
We calculate the total probability that Amber does not arrive by 9 am by considering both scenarios (bus on time and bus late):
[tex]\[
\text{Probability (not arrive by 9 am)} = (\text{Probability (bus on time)} \times \text{Probability (not arrive by 9 am if bus on time)}) + (\text{Probability (bus late)} \times \text{Probability (not arrive by 9 am if bus late)})
\][/tex]
Substituting the known values:
[tex]\[
\text{Probability (not arrive by 9 am)} = \left(\frac{5}{6} \times \frac{1}{5}\right) + \left(\frac{1}{6} \times \frac{7}{10}\right)
\][/tex]
### Step 5: Simplify the expression.
First, simplify each term separately:
[tex]\[
\frac{5}{6} \times \frac{1}{5} = \frac{5 \times 1}{6 \times 5} = \frac{1}{6}
\][/tex]
[tex]\[
\frac{1}{6} \times \frac{7}{10} = \frac{1 \times 7}{6 \times 10} = \frac{7}{60}
\][/tex]
Now, add the two results together:
[tex]\[
\frac{1}{6} + \frac{7}{60}
\][/tex]
To add these fractions, find a common denominator. The common denominator is 60:
[tex]\[
\frac{1}{6} = \frac{10}{60}
\][/tex]
So,
[tex]\[
\frac{10}{60} + \frac{7}{60} = \frac{17}{60}
\][/tex]
### Conclusion:
The probability that Amber does not arrive at work by 9 am is:
[tex]\[
\boxed{\frac{17}{60}}
\][/tex]
