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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:.

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1.5

Conditional Probability

Conditional Probability

The multiplication rule


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Definition 1.12

The conditional probability of an event A given than an

event B has already occurred is given by

Solution:

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


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arts

on time is P(D)=0.83;

and arrives on time is

given

that it departed on time,

given

that it has arrived on time.

Solution:

Example 1.13

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


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occur, then

Theorem 1.2

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

are equivalent,

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.


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proof:

Following theorem generalizes these results to n events

Theorem 1.3(The multiplication rule)

If

then


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Solution:

Example 1.14

Suppose that we have a fuse box containing 20 fuses,

which 5 are defective,

if 2 fuses are selected at random

of

and removed from the box in succession without replacing

the first,

what is the probability that both fuses aredefe-

ctive?


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Solution:

Example 1.15

One bag contains 4 white balls and 3 black balls,

and a

second bag contains 3 white balls and 5 black balls.

One

ball is drawn from the first bag and placed unseen in

thesecond bag.

What is the probability that a ball now

drawn from the second bag is black?


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then

Solution:

We denote the conditional probability that A occurs

give that B has occurred by


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Solution:

a. The probability that a plane arrives on

time

given that it departed on time is

b. The probability that a plane departed on time

given that it has arrived on time is


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Proof:


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then we interpretas the event

that A occ-

We shall let A be the event that the first

Solution:

fuse is defective and B the event that the second fuse is

defective;

urs, and then B occurs after A has

occurred.

The probability of first removing a

defective fuse is

removing a second defective

1/4;

then the probability of

Hence

fuse from the

remaining 4 is 4/19.


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the

drawing of a black ball from bag 1,

a black

ball from

61

Solution:

Let B1, B2, and W1 represent,

respectively,

bag 2,

and a white ball from bag 1.

We are interested in

the union of the

mutually exclusive

events

The various

possibilities and

their probabilities

are illustrated in

Figure 1.2 .


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