Independent Events and Conditional Probability 507- Suppose the sex of a newborn child is viewed as an experiment with two equally
likely outcomes. Assuming that each child represents an independent trial, what is the
probability of a family with four children having two, three, or four girls? Suppose the
probability of the child being a girl is 0.52; now what is the probability of two, three,
or four boys? - Find the probability of getting a five exactly twice in seven rolls of a fair die.
rnIndependent Events and Conditional Probability
Knowing that a fair coin came up heads on the first toss of a two-toss experiment does
not cause us to believe that the chance of getting heads on the second toss is other than
50%. After all, the two tosses are physically unrelated. On the other hand, we now know
that the chance of getting tails on both tosses is zero. Naturally, we anticipate that phys-
ically unrelated events resulting from, say, separate tosses of a fair coin do not affect
one another. Sometimes, events arising from a single physical experiment also behave as
though they are unrelated: Information about the occurrence of one does not shed any
light on the occurrence of the others. This phenomenon is modeled by the concept of
independence.
The subject of conditional probability tells how to revise probabilities in light of new
information. Typical examples include how to revise the probability that one of two coins
being flipped will come up heads once you know that the first coin did not. Quite a different
use of conditional probability is to determine the likelihood that the result of a medical test
result is a false-negative (that is, a positive) result. The result is a false negative (positive)
if the true result is positive (negative) but if the test gives a negative (positive) result.
First, we give the mathematical definition of what it means for two events in the same
sample space to be independent. This definition is intended to capture the notion, described
above, that some two events do not seem to influence one another. Next, we propose a way
to change the probability of an event A given that an event B in the same sample space
has occurred. This is called the conditional probability of A given B. After defining in-
dependence and conditional probability, we argue that two events in the same sample
space are independent precisely when no change is made to the probability of one event
(according to the definition of the conditional probability) if we learn that the other event
has occurred. In this way, these two mathematical definitions work together to model-and
to make precise-the notion that some pairs of events do not seem to give any informa-
tion about each other. Later, we will explore the properties and applications of conditional
probabilities.8.5.1 Independent Events
Now we give the mathematical definition intended to capture the notion that two events in
the same sample space do not influence one another. Later, we will see why this definition
does the job.
Definition 1. A pair of events A and B belonging to the same sample space are said to