Stat 104, Section 3 Handout Solutions
Benny, Thursday, 4:00pm, SC 309
Key concepts
•
Probability and Conditional Probability
•
Unions, intersections, complements, disjoint events
•
2x2 tables
•
Independence of events
•
Bayes’ Rule/Theorem
Practice Problems
1. M&M’s  sample space, basic probability and cond. prob. Playing with sets.
I have 5 M&M’s in a bag. 2 are blue, 2 are red, and 1 is yellow. I plan on eating 2 more.
a.
Write the sample space of outcomes for pairs of M&M’s that I will eat? (Here and afterwards,
order does not matter)
S
=
{
(
b
1
, b
2
)
,
(
b
1
, r
2
)
,
(
b
2
, r
1
)
,
(
b
1
, r
2
)
,
(
b
2
, r
2
)
,
(
b
1
, y
)
,
(
b
2
, y
)
,
(
r
1
, y
)
,
(
r
2
, y
)
,
(
r
1
, r
2
)
}
Note that we label M&M’s with the same color separately to account for the fact that they are
more frequently found.
b.
Write down the set of events for:
•
A: eating two mismatched M&M’s (different colors)
A
=
{
(
b
1
, r
1
)
,
(
b
2
, r
1
)
,
(
b
1
, r
2
)
,
(
b
2
, r
2
)
,
(
b
1
, y
)
,
(
b
2
, y
)
,
(
r
1
, y
)
,
(
r
2
, y
)
}
•
B: eating at least one blue M&M
B
=
{
(
b
1
, b
2
)
,
(
b
1
, r
1
)
,
(
b
2
, r
1
)
,
(
b
1
, r
2
)
,
(
b
2
, r
2
)
,
(
b
1
, y
)
,
(
b
2
, y
)
}
c.
What is the probability of A and B?
P
(
A
) =
# outcomes in A
# of outcomes in S
=

A


S

=
.
8
P
(
B
) =
# outcomes in B
# of outcomes in S
=

B


S

=
.
7
1
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d.
Are A & B disjoint?
How do you know?
No for several reasons.
In terms of their set
representation, there are outcomes that appear in both A and B. Numerically,
P
(
A
) +
P
(
B
)
>
1,
which is not possible for disjoint events. Finally, it is intuitively clear that just because one eats
two mismatched M&M’s (A) does not mean one cannot eat at least one blue M&M.
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 Fall '11
 MichaelParzen
 Conditional Probability, Probability, Probability theory, M&M

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