Rami travels to work each day, either by bus or by train. The probablility that he travels by bus is 3/5. If he travels by bus, the probablibty that he buys coffere is 2/3/ If he travels by train, the probablitliy that he buys cofffe is 3/4. Find the probablity that rami buys coffee when he travels to work

To solve this problem, we need to use the law of total probability. We'll break it down step by step.

### Step 1: Define the probabilities.
- Let \( P(B) = \frac{3}{5} \) be the probability that Rami travels by bus.
- Let \( P(T) = 1 - P(B) = \frac{2}{5} \) be the probability that Rami travels by train.

- Let \( P(C|B) = \frac{2}{3} \) be the probability that Rami buys coffee given that he travels by bus.
- Let \( P(C|T) = \frac{3}{4} \) be the probability that Rami buys coffee given that he travels by train.

### Step 2: Calculate the total probability that Rami buys coffee.
We want to find \( P(C) \), the total probability that Rami buys coffee. According to the law of total probability:

\[
P(C) = P(C|B) \times P(B) + P(C|T) \times P(T)
\]

### Step 3: Substitute the values and compute.

\[
P(C) = \left(\frac{2}{3} \times \frac{3}{5}\right) + \left(\frac{3}{4} \times \frac{2}{5}\right)
\]

First, compute each term:

\[
\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}
\]

\[
\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}
\]

Now, add the two results:

\[
P(C) = \frac{2}{5} + \frac{3}{10}
\]

Convert \( \frac{2}{5} \) to a fraction with denominator 10:

\[
\frac{2}{5} = \frac{4}{10}
\]

So,

\[
P(C) = \frac{4}{10} + \frac{3}{10} = \frac{7}{10}
\]

### Final Answer:
The probability that Rami buys coffee when he travels to work is \( \frac{7}{10} \).

Do you want details on any part of this solution or have any questions?  

Here are some related questions:

1. What is the probability that Rami travels by train and buys coffee?
2. What is the probability that Rami travels by bus and does not buy coffee?
3. How would the probability change if the probability of buying coffee on the bus were higher?
4. What is the probability that Rami does not buy coffee regardless of the mode of transportation?
5. How does the law of total probability help in solving such problems?

**Tip:** The law of total probability is useful when dealing with scenarios that can be divided into multiple cases, each with its own conditional probability.

Rami travels to work each day, either by bus or by train. The probability that he travels by bus is 3/5. If he travels by bus, the probability that he buys coffee is 2/3. If he travels by train, the probability that he buys coffee is 3/4. Find the probability that Rami buys coffee when he travels to work.

Discover how to calculate the probability that Rami buys coffee when traveling to work, using the law of total probability. This problem involves conditional probabilities, including the chances of taking the bus or train. Suitable for students in Grades 9-12.

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