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40 CHAPTER 2 PROBABILITY

What is the conditional probability that b is selected second given that a is selected
first? Because this question considers the results of each pick, the ordered sample space is
used. The definition of conditional probability is applied as follows. Let E denote the
1
event that the first item selected is a and let E 2 denote the event that the second item se-
lected is b. Then,

E 5ab, ac6 and E 5ab, cb6 and E ¨ E 5ab6
1
2
1
2
and from the definition of conditional probability
1
6
P1E 2 ƒ E 1 2 P1E 1 ¨ E 2 2
P1E 1 2 1
2
1
3

When the sample space is larger, an alternative calculation is usually more convenient.
For example, suppose that a batch contains 10 parts from tool 1 and 40 parts from tool 2. If
two parts are selected randomly, without replacement, what is the conditional probability that
a part from tool 2 is selected second given that a part from tool 1 is selected first? There are
50 possible parts to select first and 49 to select second. Therefore, the (ordered) sample space
has 50 49 2450 outcomes. Let E 1 denote the event that the first part is from tool 1 and E 2
denote the event that the second part is from tool 2. As above, a count of the number of out-
comes in E 1 and the intersection is needed.
Although the answer can be determined from this start, this type of question can be
answered more easily with the following result.

To select randomly implies that at each step of the sample, the items that remain in
the batch are equally likely to be selected.

If a part from tool 1 were selected with the first pick, 49 items would remain, 9 from tool 1 and
40 from tool 2, and they would be equally likely to be picked. Therefore, the probability that
a part from tool 2 would be selected with the second pick given this first pick is

P1E ƒ E 2 40
49.
1
2
In this manner, other probabilities can also be simplified. For example, let the event E
consist of the outcomes with the first selected part from tool 1 and the second part from tool 2.
To determine the probability of E, consider each step. The probability that a part from tool 1
is selected with the first pick is P1E 2 10
50 . The conditional probability that a part from
1
tool 2 is selected with the second pick, given that a part from tool 1 is selected first is
P1E ƒ E 2 40
49 . Therefore,
1
2
40 10
P1E2 P 1E 2 0 E 1 2P1E 1 2 0.163
49 50