If you take a bus to work in the morning there is a 20% chance you’ll arrive late. When you go by bicycle there is a 10% chance you’ll be late. 70% of the time you go by bike, and 30% by bus. Given that you arrive late, what is the probability you took the bus?
P(took the bus | being late) = P(took the bus and being late) / P(being late)
How can I solve for the bolded part?
Good question well first you gotta take the bus $(0.3)$ then you need to be late on the bus $(0.2)$ so your bolded probability is $0.3 \times 0.2$ otherwise known as $0.06$ and since I am here that means the answer is $0.06/(0.06+0.07)=6/13 = 0.46$ or just under half.
Does this seem plausible? Well taking the bike is generally more likely than the bus but the bus is less reliable (you are twice as likely to be late) so if we know we are late then that increases the odds that we took the bus and yeah this number passes the sniff-test.