Figure 1 shows how 25 people travelled to work - Edexcel - A-Level Maths Statistics - Question 4 - 2012 - Paper 2

To determine if events B and T (bicycle and train) are independent, we can check if:

$P(B ext{ and } T) = P(B) imes P(T)$

From the diagram, we note that:

• The total number of people is 25.
• The number of people who travel by bicycle only is 4, by train only is 3, and those who use both are 5.

So, we can calculate:

• $P(B) = \frac{4 + 5}{25} = \frac{9}{25}$
• $P(T) = \frac{3 + 5}{25} = \frac{8}{25}$
• $P(B ext{ and } T) = \frac{5}{25}$

Calculating:

$P(B ext{ and } T) = \frac{5}{25} \neq \left(\frac{9}{25} \times \frac{8}{25}\right) \neq \frac{72}{625}$

Thus, B and T are not independent.

The probability that a person walks to work can be calculated as:

$P(W) = \frac{7}{25} = 0.28$

The probability that a person travels to work by both bicycle and train can be calculated as:

$P(B \text{ and } T) = \frac{5}{25} = 0.2$

To find the conditional probability that a person takes the train given that they travel by bicycle, we utilize:

$P(T | B) = \frac{P(B \text{ and } T)}{P(B)}$

Where:

• $P(B \text{ and } T) = \frac{5}{25}$
• $P(B) = \frac{9}{25}$

Thus,

$P(T | B) = \frac{5/25}{9/25} = \frac{5}{9} \approx 0.555$