Probability Distributions - Oxford University Press

182 Business Statistics Using Excel
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Independent events Two
events are independent
if the occurrence of one
of the events has no influence
on the occurrence of
the other event.
n{A or B} = n{A} + n{B} 2 n{A > B}. Consequently by transforming the events into probabilities
the general addition law is as follows:
P(A or B) = P(A) + P(B) 2 P(A > B) (5.3)
Thus if two events are mutually exclusive, P(A > B) = 0.
Example 5.2
A card is chosen from an ordinary
pack of cards. Write down the probabilities
that the card is: (a) black and
an ace, (b) black or an ace, and (c) neither
black nor an ace.
Let event A and B represent the
events obtaining an ace card and B a
black card respectively. The sample
space is represented by Figure 5.3.
Number of outcomes in A∩B 2
(a) P(B and A) =
Total numberofoutcomes
52 0.0385
= =
(b) P(B or A) = P(B) + P(A) – P(B > A) = 26 4 2 28
+ − = = 0. 538462
52 52 52 52
(c) P(neither B nor A) = 1 – P(B or A) = 1 – 0.5385 = 0.4615
Let’s look at the general multiplication law. We mentioned above mutually exclusive events,
i.e. events that cannot occur at the same time, but what about completely independent
events? An example is rolling a die twice. The fact that we got 6 on the frst roll, for example,
cannot influence the outcome of the second roll. Similarly take the example of picking a
ball from a bag, if it were replaced before another was picked nothing changes; the sample
space remains the same. Drawing the frst ball and replacing it cannot affect the outcome
of the next selection. In these examples we have the notion of independent events. If two
(or more) events are independent then the general multiplication law applies:
P(A > B) = P(A) * P(B) (5.4)
Note The terms independent and mutually exclusive are different and apply to
different things. If A and B are events with non-zero probabilities, then we can show that
P(A > B):
• P(A > B) = 0, if mutually exclusive. Mutually exclusive events cannot occur at the same
time.
• P(A > B) ≠ 0, if independent. Independent events do not influence each other.
A
2
Figure 5.3
2
24
B
24