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August STAT/MA 31, 2011
Problem Set 5
Name:
1. Waking up at random.
On each weekday, a student wakes up before 8 AM with probability .65, or after 8 AM
with probability .35.
On each weekend, a student wakes up before 8 AM with probability .22, or after 8 AM
with probability .78.
Suppose that a student wakes up and cant remember which day it is!
(a.) If the student sees that it is before 8 AM, what is the probability it is a weekday?
(b.) If the student sees that it is after 8 AM, what is the probability it is a weekday?
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2. Socks. A matching pair of blue socks, a matching pair of red socks, and one lone white
right-footed sock are in a drawer.
Suppose that a person blindly pulls a sock out of the drawer and it is not red (i.e., it is
either the white sock or one of the blue socks). He discards this sock, so four socks remain.
Now he reaches into the drawer again and chooses a sock at random from the four socks
that remain. What is the probability that this sock is blue?
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3. Pair of dice. Roll a blue die and red a die. Given that the blue die has an odd value,
which is the probability that the sum of the two dice is exactly 4?
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4. Pair of dice. Roll a blue die and a red die. Given that the blue die has a value of 4 or
smaller, what is the probability that the sum of the two dice is 7 or larger?
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5. Coin ips and then dice. Claire ips a coin until she gets heads for the rst time.
Say it takes her n times. Then (afterwards) she rolls exactly n dice. What is the probability
that none of the dice show the value 1?
(For instance, if it takes her 7 ips to get heads for the rst time, then she rolls 7 dice.
Hint: It is enough to use Remark 5.7 at the bottom of page 77. You do not need Bayes
Theorem itself for this problem.)
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6. Design your own problem and solution. Create your own problem about Bayes
Theorem. Design your problem in such a way that you would nd it enjoyable and also
interesting (i.e., not completely trivial) if you found this problem in a probability book.
Please provide the answer for your problem too.
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