Often we are interested in the probabilities of various combinations of events.
Suppose two events A and B are independent (that is, the occurance of one event will not affect the occurance of the other). Then P(A and B) = P(A)P(B).
Example: Roll a 4-sided die and a 6-sided die. Let A be the event that the 4-sided die gives a 3 and let B be the event that the 6-sided die gives a 5. The event A and B is the event that the 4-sided die gives a 3 and the 6-sided die gives a 5. Then P(A) = 1/4, P(B) = 1/6 and P(A and B) = 1/24. See the spreadsheet DiceIndep.xls to see how this works.
Example: Earthquake faults in the Bay area of California. A panel of experts in 1990 determined that there are four major faults in the Bay area that could experience magnitude 7 earthquakes within the next 30 years. Each fault was judged to have a probability of 0.25 to have such an earthquake in that time frame, and each fault is roughly independent of the others. So each fault has a 0.75 probability of not having a major earthquake in that time frame. So the probability of no major earthquakes on all four faults is about 0.75 multiplied by itself 4 times, or 0.32. So the probability of the complementary event (the probability of at least one quake on one of the faults) is 1 - 0.32 = 0.68.
Example: De Mere's "scandal". He believed that one must roll two dice 24 times before the probability is at least 0.5 of getting at least one double-six (a six on both dice). But he knew from experience that at least 25 rolls is necessary before the probability is at least 0.5. He consulted his friends Pascal and Fermat for an explanation. The truth of the matter is as follows: The probability of no double-six on one roll is 35/36. So the probability of no double-six on n rolls is (35/36)n. Therefore the complementary event (at least one double-six in n rolls) is 1 - (35/36)n. If n were 24, this would be 0.49; if n were 25, this would be 0.505.
Now we consider the probability of the event A or B. This is the probability that either A occurs or B occurs or they both occur. [This usage of the word or is called the "inclusive or".] The relevant formula is: P(A or B) = P(A) + P(B) - P(A and B).
Example: Draw a single card from a deck. Suppose A is the event that the card is a king. Suppose B is the event that the card is a heart. We have P(A) = 4/52 since there are 4 kings in a deck of 52 cards. We have P(B) = 13/52 since there are 13 hearts in a deck of 52 cards. Then the probability that the card is a king or a heart or both is P(A or B) = P(A) + P(B) - P(A and B) = 4/52 + 13/52 - 1/52, since there is exactly one king of hearts in the deck. We see from the arithmetic what's going on here: We're trying to find the probability that a card is a king or a heart. There are 4 kings and 13 hearts from 52 cards, so we're tempted to say the probability is 17/52. But we've accidently counted one card twice: the king of hearts. This is why the term P(A and B) in the formula is necessary.
Example: Draw a single card from a deck. Let A
be the event that the card is a king, but now let B be the event
that the card is a jack. We see P(A) = P(B)
= 4/52, and P(A and B) = 0, so P(A
or B) = P(A) + P(B) = 8/52.
When P(A and B) = 0, that is, A and B
cannot both occur, we say the events are "mutually exclusive".