A train arrives at a station and waits 3.5 minutes before departing. Another train arrives at the station 16 minutes later, repeating the cycle. Identify the probability that a train will not be at the station when you arrive. 0.82 0.84 0.22 0.18

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UpStudy AI · Accepted Answer

To solve this problem, we need to analyze the timing of the trains and the intervals during which they are at the station.

1. **Understanding the Train Schedule:**
   - The first train arrives and waits for 3.5 minutes.
   - After it departs, another train arrives 16 minutes later.
   - Therefore, the total cycle time from the arrival of the first train to the arrival of the second train is $$3.5 + 16 = 19.5$$ minutes.

2. **Identifying the Time Intervals:**
   - During the first 3.5 minutes, the first train is at the station.
   - After the first train leaves, there are 16 minutes until the next train arrives, during which no train is at the station.

3. **Calculating the Probability:**
   - The total time in the cycle is 19.5 minutes.
   - The time when no train is at the station is 16 minutes.
   - The probability that a train will not be at the station when you arrive can be calculated as the ratio of the time without a train to the total cycle time.

$$
   P(\text{no train}) = \frac{\text{time without a train}}{\text{total cycle time}} = \frac{16}{19.5}
   $$

Now, let's calculate this probability.

$$
P(\text{no train}) = \frac{16}{19.5}
$$

I'll perform the calculation now.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{16}{19.5}$$
- step1: Convert the expressions:
  $$\frac{16}{\frac{39}{2}}$$
- step2: Multiply by the reciprocal:
  $$16\times \frac{2}{39}$$
- step3: Multiply:
  $$\frac{16\times 2}{39}$$
- step4: Multiply:
  $$\frac{32}{39}$$
The probability that a train will not be at the station when you arrive is approximately $$0.8205$$, which can also be expressed as $$\frac{32}{39}$$.

Given the options:
- 0.82
- 0.84
- 0.22
- 0.18

The closest answer is **0.82**.

Thus, the probability that a train will not be at the station when you arrive is **0.82**.
The probability that a train will not be at the station when you arrive is **0.82**.

A train arrives at a station and waits 3.5 minutes before departing. Another train arrives at the station 16 minutes later, repeating the cycle. Identify the probability that a train will not be at the station when you arrive. 0.82 0.84 0.22 0.18

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