What are independent eventsin probability?
In probability theory, events are independent if the probability that they occur is equal
to the product (multiply together) of the probabilities of either two or more individual
events. (This is also often called statistical independence.) In addition, the occurrence
of one of the events can give no information about whether or not the other event(s)
will occur; that is, the events have no influence on each other.
For example, two events, Aand B,are independent if the probability of both
occurring equals the product of their probabilities, or P (A) | P(B) (the symbol “|” is
often used to depict the product of the events in probability theory). One good exam-
ple involves playing cards. If we wanted to know the probability of two people each
drawing a king of diamonds (two independent events), it would be defined as A1/52
(the probability that one person will draw a king of diamonds) and B1/52 (the prob-
ability that the other person will draw a king of diamonds, assuming the first person
puts the first drawn card back into the deck). Substituting the numbers into the equa-
tion, the result is: 1/52 | 1/52 0.00037, or a slight chance that both people will draw
the king of diamonds from the deck.What is Bayes’s theorem?
Bayes’s theorem is a result that lets new information be used to update the conditional
probability of an event. The theorem was first derived by English mathematician
Thomas Bayes (1702–1761), who developed the concept to use in situations in which
250 probability can’t be directly calculated. The theorem gives the probability that a cer-
What is conditional probability?
C
onditional probability is often phrased as “event Aoccurs given that event B
has occurred.” The common notation is a vertical line, or A| B(said as “A
given B”). Thus, P (A| B) denotes the probability that event Awill occur given
that event Bhas occurred already.Since there is always room for improvement—even in probability—condi-
tional probability incorporates the idea that once more information becomes
available, the probability of further outcomes can be revised. For example, if a
person brings a car in for an oil change every 3,000 miles, it can be calculated
that there is a probability of 0.9 that the service on the car will be completed
within two hours. But if the car is brought in during a seatbelt recall, the proba-
bility of getting the car back in two hours might be reduced to 0.6. This is the
conditional probability of getting the car back in two hours if there is a seatbelt
recall taking place.