The probability that Amber's bus arrives on time is [tex]\frac{5}{6}[/tex].
If her bus is on time, then the probability she arrives at work by 9 am is [tex]\frac{4}{5}[/tex].

If her bus is late, then the probability she arrives at work by 9 am is [tex]\frac{3}{10}[/tex].

Work out the probability that Amber does not arrive at work by 9 am.

Give your answer as a fraction in its simplest form.



Answer :

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]

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