P(event happens) + P(event doesn't happen) = 1.neilcao wrote:Two independent events, the probability of any of them to occur is P, what's the probability that one happens and the other not. (expressed in terms of p)
Could this be 2p（1-p）? Why is that?
If p = P(event happens), then 1-p = P(event doesn't happen).
Let's say p = P(rain on any given day).
When we want multiple events to happen together, we multiply the probabilities: P(A and B) = P(A) * P(B).
We multiply the fractions because the more events we want to happen together, the smaller the probability, and when we multiply fractions, the result just keeps getting smaller.
P(rain on the first day and no rain on the second day) = p*(1-p)
P(no rain on the first day and rain on the second day) = (1-p)*p
Either of the above will result in a favorable outcome of one day of rain. When we want one event or another event to happen, we add the probabilities: P(A or B) = P(A) + P(B).
We add the fractions because either combination of events will generate a favorable outcome of one day of rain, thereby increasing the probability that we'll get what we want. When we add fractions, the result gets bigger, reflecting the increasing probability.
Adding the results above, we get:
p(1-p) + (1-p)p = 2p(1-p).
