Guided Example #2 - Conditional Probability

Tony takes either a bus or the subway to visit his uncle with probabilities 0.3 and 0.7, respectively. When he takes the bus, he is late 30% of the days. When he takes the subway, he is late 20% of the days. If Tony is late for visit on a particular day, what is the probability that he took the bus?

A. $10 %$ B. $30 %$ C. $3 %$ D. $9 %$

Step 1

Define the following events:
B:Tony takes the bus S:Tony takes the subway L:Tony is late

Step 2

It is given that $P (B) = 0.3, P (S) = 0.7$ ,and $P (L) = 0.2$ .

Step 3

Therefore,
$P (L | B) = \frac{P (L ⋂ B)}{P (B)} \Rightarrow 0.3$
$= \frac{P (L ⋂ B)}{0.3} \Rightarrow P (L ⋂ B)$
$= 0.009 = 9 %$

All Steps

Define the following events:
B:Tony takes the bus S:Tony takes the subway L:Tony is late

It is given that $P (B) = 0.3, P (S) = 0.7$ ,and $P (L) = 0.2$ .

Therefore,
$P (L | B) = \frac{P (L ⋂ B)}{P (B)} \Rightarrow 0.3$
$= \frac{P (L ⋂ B)}{0.3} \Rightarrow P (L ⋂ B)$
$= 0.009 = 9 %$