MAT 230 EXAM TWO
This document is proprietary to Southern New Hampshire University. It and the problems within
may not be posted on any nonSNHU website.
Keith Ellison
1
Directions: Type your solutions into this document and be sure to show all steps for arriving at
your solution. Just giving a final number may not receive full credit.
Problem 1
This question has 2 parts.
Part 1:
Suppose that
F
and
X
are events from a common sample space with
P
(
F
)
6
= 0 and
P
(
X
)
6
= 0.
(a) Prove that
P
(
X
) =
P
(
X

F
)
P
(
F
) +
P
(
X

¯
F
)
P
(
¯
F
).
Hint:
Explain why
P
(
X

F
)
P
(
F
) =
P
(
X
∩
F
) is another way of writing the definition of conditional probability, and then use
that with the logic from the proof of Theorem 4.1.1.
From the conditional probability,
P(X—F)=P(XF)/P(F) P(X—F)P(F)=P(XF) P(X—F)=P(XF’)/P(F) P(X—F)P(F)=P(XF’)
P(X)=P(X) P(X)=P(X(FF)) =P((XF)(XF))
the probability is independent,
P((XF)(XF))=P(XF)+P(XF) =P(X)P(F)+P(X)P(F) P(X)=P(XF)+P(XF)
(b) Explain why
P
(
F

X
) =
P
(
X

F
)
P
(
F
)
/P
(
X
) is another way of stating Theorem 4.2.1 Bayes’
Theorem.
From the conditional density function,
P(F—X)=P(XF)/P(X)
P(X—F)P(F)=P(XF) Used in the conditional equation, P(F—X)=P(XF)/P(X) P(F—X)
=P(X—F)P(F)/P(X)
Part 2:
A website reports that 70% of its users are from outside a certain country. Out of their users
from outside the country, 60% of them log on every day. Out of their users from inside the country,
80% of them log on every day.
(a) What percent of all users log on every day? Hint: Use the equation from Part 1 (a).
Use ’X’ as the total percentage of users that log on everyday F is the percentage of users
from outside the country who log on everyday F
F
be the percentage of users from inside
the country logging on everyday. The probability of the total percentage of users logging on
everyday; P(X)=100x
Probability of users log from outside the country; P(F)=70x
Probability of users log from inside the country; P(
F
)=30x
Probability of users log from outside the country everyday; P(X—F)P(F)=60/100(70x)=42x
Probability of users log from inside the country everyday, P(X—
F
)P(
F
)=80/100(30x)=24x
Total probability of users logging on everyday is given by using the proof of Part 1(a),
P(X)=P(X—F)P(F)+P(X—
F
)P(
F
) P(X)=42x+24x=66x P(X)=66x.