Use the table to answer the question. Train Arrival Train Path On Time Arrival Late Arrival Total In the city 54 6 60 City to city 27 3 30 Total 81 9 90 Let event A be the train that is on time and event B be the city to city train. What is P(A and B) ?

Answer 1

To find $P(A\text{ and }B)$, which is the probability that a train is both on time and a city-to-city train, we need to look for the intersection of the two events:

• Event A (On Time): 81 trains are on time (54 in the city + 27 city to city)
• Event B (City to City): 30 trains are city to city (27 on time + 3 late)

From the table, we see that:

• The number of city-to-city trains that are on time is 27.

The total number of trains is 90. Therefore, the probability $P(A\text{ and }B)$ is calculated as:

$$P(A\text{ and }B)=\frac{\text{Number of city to city trains that are on time}}{\text{Total number of trains}}=\frac{27}{90}$$

Now, simplifying:

$$P(A\text{ and }B)=\frac{27}{90}=\frac{3}{10}=0.3$$

Thus,

$$P(A\text{ and }B)=0.3$$