Probability Tom is late if he goes to work by bus or by car

Question

Tom walks to bus stop 4/5 days. The probability he catches it is 19/20. If he misses it, he's late for work. 1/5 days his girlfriend drives him to work. Her car breaks down at 1/4 journeys resulting Tom being late for work. 

A) what's the probability that Tom is late for work on any random day? 

B)  given he arrives on time on any random day, what's the probability he came on the bus? 

 

I have calculated it, but I'm not sure for the results, as I think I didn't get the events right. 

Answer 1

A) Break down the situations as follows:

 

Bus Stop: (4/5)

  - Catches: (19/20)

  - Misses: (1/20)

 

Dropped off: (1/5)

  - Breaks down: (1/4)

  - Makes it: (3/4)

 

p(late) = p(bus and miss) + p(dropped off and breaks down) = (4/5)*(1/20) + (1/5)*(1/4) = 4/100 + 1/20 = 0.09

 

B) Given that he arrives, p(bus)

 

p(bus | arrival) = p(bus and arrives) / p(arrives)

 

p(bus and arrives) = (4/5)*(19/20) = 76/100 = 0.76

p(arrives) = p(bus and arrives) + p(dropped off and arrives) = 76/100 + (1/5)*(3/4) = 91/100

 

p(bus | arrives) = (76/100)/(91/100) = 0.835