Probability
In probability theory, events E1, E2, ..., En are said to be mutually exclusive if the occurrence of any one of them automatically implies the non-occurrence of the remaining n − 1 events. Therefore, two mutually exclusive events cannot both occur. Formally said, the intersection of each two of them is empty (the null event): A and B = ∅. In consequence, mutually exclusive events have the property: P(A and B) = 0.
For example, one cannot draw a card that is both red and a club because clubs are always black. If one draws just one card from the deck, either a red card or a club can be drawn. When A and B are mutually exclusive, P(A or B) = P(A) + P(B). One might ask, "What is the probability of drawing a red card or a club?" This problem would be solved by adding together the probability of drawing a red card and the probability of drawing a club. In a standard 52-card deck, there are twenty-six red cards and thirteen clubs: 26/52 + 13/52 = 39/52 or 3/4.
One would have to draw at least two cards in order to draw both a red card and a club. The probability of doing so in two draws would depend on whether the first card drawn were replaced before the second drawing, since without replacement there would be one fewer card after the first card was drawn. The probabilities of the individual events (red, and club) would be multiplied rather than added. The probability of drawing a red and a club in two drawings without replacement would be 26/52 * 13/51 = 338/2652, or 13/102. With replacement, the probability would be 26/52 * 13/52 = 338/2704, or 13/104.
In probability theory the word "or" allows for the possibility of both events happening. The probability of one or both events occurring is denoted P(A or B) and in general it equals P(A) + P(B) – P(A and B). Therefore, if one asks, "What is the probability of drawing a red card or a king?", drawing any of a red king, a red non-king, or a black king is considered a success. In a standard 52-card deck, there are twenty-six red cards and four kings, two of which are red, so the probability of drawing a red or a king is 26/52 + 4/52 – 2/52 = 28/52. However, with mutually exclusive events the last term in the formula, – P(A and B), is zero, so the formula simplifies to the one given in the previous paragraph.
Events are collectively exhaustive if all the possibilities for outcomes are exhausted by those possible events, so at least one of those outcomes must occur. The probability that at least one of the events will occur is equal to 1. For example, there are theoretically only two possibilities for flipping a coin. Flipping a head and flipping a tail are collectively exhaustive events, and there is a probability of 1 of flipping either a head or a tail. Events can be both mutually exclusive and collectively exhaustive. In the case of flipping a coin, flipping a head and flipping a tail are also mutually exclusive events. Both outcomes cannot occur for a single trial (i.e., when a coin is flipped only once). The probability of flipping a head and the probability of flipping a tail can be added to yield a probability of 1: 1/2 + 1/2 =1.
Read more about this topic: Mutually Exclusive Events
Famous quotes containing the word probability:
“Legends of prediction are common throughout the whole Household of Man. Gods speak, spirits speak, computers speak. Oracular ambiguity or statistical probability provides loopholes, and discrepancies are expunged by Faith.”
—Ursula K. Le Guin (b. 1929)
“Only in Britain could it be thought a defect to be “too clever by half.” The probability is that too many people are too stupid by three-quarters.”
—John Major (b. 1943)
“The probability of learning something unusual from a newspaper is far greater than that of experiencing it; in other words, it is in the realm of the abstract that the more important things happen in these times, and it is the unimportant that happens in real life.”
—Robert Musil (1880–1942)