When you come to think about probability, it plays a larger role than most of us realize. Winning a lottery, birthday attacks, filtering spams, identifying intrusions, authenticating using biometrics and coincidences are some of the immediate ones that come into my numerically-tuned ;) mind. Sometime our intuitive feeling make a guess about something happening but we often find that mathematical implications are quite different from what we have thought of. Wikipedia has a nice example on false-positives which provides evidence for the above claim.
I found this book : Understanding Probability: Chance Rules in Everyday Life by Henk C. Tijms really interesting for two reasons. First, I like the catchy title. Second is a more serious one: throughout the book, the author beautifully shows how probability is an intricate part of our life.
For those who want to stretch their minds, here's the infamous "The Monte Hall Problem" if you have not come across this one already.
Suppose you’re on a game show, and you’re given the choice of three doors:
Behind one door is a car; there's nothing behind the other two doors. You pick a door, say number 3, and the host, who knows what’s behind the doors, opens another door, say number 2, which is empty. Then you are given the opportunity to select the remaining door (number 1 in this case) or stick to your fist choice.
Does it matter if you change your mind and pick the first door? Surprisingly the answer is yes. Intuitively you, like me, may believe you have the probability of 0.5 (1 in 2 chances) of winning, but it turns out that it is not the case.
What is the probability of winning the car if you stick with your first choice? 1/3 (This should be simple enough to understand, as it is the choice of one in three)
What is the probability of winning the car if you change your mind and choose the first door? 2/3 (You only have two choices, stick with the first one, 1/3, or change your mind 1-1/3)
You see, second thoughts are not a bad thing given certain conditions change during the cause!