Why P(A|B) isn't equal to P(B|A)?

Question

I don't understand.
Could somebody please explain it using an easy-to-understand example?
Thank you.

Answer 1

OK, here's a simple example: I (almost) always carry an umbrella in my briefcase during the wetter months of the year, so I don't have to check the forecast. I don't bother during summer, normally.

So let's divide the year into winter and summer and say that it rains half the time in winter, and I always have my umbrella, but in summer it only rains one day in 20 and I never carry my umbrella. Let's also assume the year is 360 days long to make the sums easier. ;-)

So we have
winter, raining, umbrella: 90 days
winter, not raining, umbrella: 90 days
summer, raining, no umbrella: 9 days
summer, not raining, no umbrella: 171 days

P(rains) = 99 / 360, P(umbrella) = 1/2.

P(rains | umbrella) = 90 / (90+90) = 1/2
P(umbrella | rains) = 90 / (90 + 9) = 10/11.

In other words, I'll almost certainly have my umbrella if it's raining (because it basically only rains in winter), but just because I have my umbrella doesn't make it especially likely to rain.

Answer 2

P(A|B) is the probability of getting situation A given the condition B, while P(B|A) is the probability of getting situation B given the condition A,

Answer 3

It would only be true if B = A and the probability of this is highly unlikely.

I almost think I made a joke there......

Answer 4

P(A/B) is not equal to P(B/A)
P(A/B)=P(A intersection B)/P(B)
P(B/A)=P(B intersection A)/P(A)