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STAT 400 – BL1
Examples for 08/30/2011
Fall 2011
Events
A
and
B
are
independent
if
and
only
if
P
(
B
A
)
=
P
(
B
)
P
(
A
B
)
=
P
(
A
)
P
(
A
∩
B
)
=
P
(
A
)
⋅
P
(
B
)
Note that if two events, A and B, are mutually exclusive, then P( A
∩
B ) = 0.
Therefore, two
mutually exclusive events cannot be independent, unless at least one of them has probability 0.
1.
The probability that a randomly selected student at Anytown College owns a bicycle is
0.55, the probability that a student owns a car is 0.30, and the probability that a student
owns both is 0.10.
Are events {a student owns a bicycle} and {a student owns a car}
independent?
P( B
∩
C )
≠
P( B )
×
P( C ).
0.10
≠
0.55
×
0.30.
B and C are
NOT independent
.
1
½
.
During the first week of the semester, 80% of customers at a local convenience store bought
either beer or potato chips (or both).
60% bought potato chips.
30% of the customers bought
both beer and potato chips.
Are events {a randomly selected customer bought potato chips} and
{a randomly selected customer bought beer} independent?
[ Recall that P(Beer) = 0.50. ]
P( B
∩
PC ) = P( B )
×
P( PC ).
0.30 = 0.50
×
0.60.
B and PC are
independent
.

This
** preview**
has intentionally

Events
A,
B
and
C
are
independent
if
and
only
if
P
(
A
∩
B
) = P
(
A
)
⋅
P
(
B
),
P
(
A
∩
C
) = P
(
A
)
⋅
P
(
C
),
P
(
B
∩
C
) = P
(
B
)
⋅
P
(
C
),
and
P
(
A
∩
B
∩
C
) = P
(
A
)
⋅
P
(
B
)
⋅
P
(
C
)
1
¾
.
Suppose that a fair coin is tossed twice.
Consider
A = {H on the first toss},
B = {H on the second toss} and
C = {exactly one H in two tosses}.
S = { TT, TH, HT, HH },
A = { HT, HH },
B = { TH, HH },
C = { TH, HT }.
P(
A ) =
1
/
2
,
P(
B ) =
1
/
2
,
P(
C ) =
1
/
2
.
a)

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