1.5. Conditional Probability. Conditional Probability. The multiplication rule. Definition 1.12. The conditional probability of an event A given than an. event B has already occurred is given by. Solution:.
The multiplication rule
The conditional probability of an event A given than an
event B has already occurred is given by
Suppose that we roll a fair six-sided die and note the score obtained. Let A = the event that the outcome is > 3 and B= the event that the outcomeis an even number. What is the conditional probability that B occurs given that A has occurred?
provided that P(B)>0
on time is P(D)=0.83;
and arrives on time is
that it departed on time,
that it has arrived on time.
The probability that a regularly scheduled flight dep-
the probability that it arrives on
and the probability that it departs
time is P(A)=0.82;
Find the probability that a plane (a) arrives on time
and (b) departed on time
If in an experiment the events A and B can both
Thus the probability that both A and B occur is equal to
the probability that A occurs multiplied by the probability
that B occurs,
given that A occurs. Since the events
it follows from Theorem
1.2 that we can also write
In other words, it does not matter which event is referred
to as A and which event is referred to as B.
Following theorem generalizes these results to n events
Theorem 1.3(The multiplication rule)
Suppose that we have a fuse box containing 20 fuses,
which 5 are defective,
if 2 fuses are selected at random
and removed from the box in succession without replacing
what is the probability that both fuses aredefe-
One bag contains 4 white balls and 3 black balls,
second bag contains 3 white balls and 5 black balls.
ball is drawn from the first bag and placed unseen in
What is the probability that a ball now
drawn from the second bag is black?
We denote the conditional probability that A occurs
give that B has occurred by
a. The probability that a plane arrives on
given that it departed on time is
b. The probability that a plane departed on time
given that it has arrived on time is
then we interpretas the event
that A occ-
We shall let A be the event that the first
fuse is defective and B the event that the second fuse is
urs, and then B occurs after A has
The probability of first removing a
defective fuse is
removing a second defective
then the probability of
fuse from the
remaining 4 is 4/19.
drawing of a black ball from bag 1,
Let B1, B2, and W1 represent,
and a white ball from bag 1.
We are interested in
the union of the
are illustrated in
Figure 1.2 .