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Sets, Combinatorics, and Probability
94. Two fair dice are rolled. The sample space S contains the 36 combinations of two numbers. For each member (r, s) of S, the random variable X(r, s) = r + s.
a. Write a table showing the values for X and the probability of those values; instead of 36 columns each
with probability 1/36, do a column for each distinct value of X and show the probability of that value.
b. Find the expected value of the sum of the numbers that come up when two fair dice are rolled.
c. Find the expected value of the sum of the numbers that come up when two fair dice are rolled. This
time let the sample space S consist of the ordered pairs (r, s) that can appear on the two dice. Use two
different random variables over this sample space, where X 1 = the value of the first component of the
ordered pair and X 2 = the value of the second component of the ordered pair. Make use of the linearity
property from Equation (4).
95. At a gambling casino, a ball will be drawn from a bin containing 43 red balls, 27 green balls, and 8 blue
balls. A player marks a game card with the color he or she believes will be picked. The prize money for
guessing the correct color is
Red
$3.00
Green
$6.00
Blue
$10.00
The price of the game card is $5.00. Find the expected value of the prize money.
96. A directory on a computer’s hard disk contains 12 files, 3 of which have viruses. If a file with a virus is
selected, the virus is detected and another file is then selected. Find the expected number of files that must
be selected in order to get a virus-free file.
97. Bit strings are sent across a computer network in packets of length 10. The probability of a bit getting corrupted (that is, a 0 gets changed to a 1 or vice versa) is 0.01. These bit errors are independent.
a. Find the probability that in a single packet there are no bit errors. (Hint: It’s OK to call a bit error a
“success.”)
b. Find the probability that there are no more than two bit errors.
c. Find the probability that there is at least one bit error.
98. Of the items produced in a certain manufacturing facility, 5% are defective. If 8 items are chosen at random, find the probability that
a. 1 is defective.
b. 2 are defective.
c. none is defective.
d. at least 1 is defective.
e. at most 1 is defective.
99. A student has failed to study for a true−false test and guesses at every one of the 10 questions. If the passing grade is 8 correct answers, what is the probability that the student will pass the quiz?
100. A baseball player has a probability p = 0.04 of hitting a home run for each at bat. Find the minimum
number of at bats he must take so that there is at least an 80% probability of hitting a home run (that is, at
least 1 home run).
101. Find the average number of comparisons to search for a target element x using the sequential search algorithm under the assumption that x is equally likely to be at any of the n positions in the list or not in the
list.
102. Find the average number of comparisons to search for a target element x using the sequential search algorithm under the assumption that x is not in the list 80% of the time, but if x is in the list it is equally likely
to be at any of the n positions.
Sets, Combinatorics, and Probability
94. Two fair dice are rolled. The sample space S contains the 36 combinations of two numbers. For each member (r, s) of S, the random variable X(r, s) = r + s.
a. Write a table showing the values for X and the probability of those values; instead of 36 columns each
with probability 1/36, do a column for each distinct value of X and show the probability of that value.
b. Find the expected value of the sum of the numbers that come up when two fair dice are rolled.
c. Find the expected value of the sum of the numbers that come up when two fair dice are rolled. This
time let the sample space S consist of the ordered pairs (r, s) that can appear on the two dice. Use two
different random variables over this sample space, where X 1 = the value of the first component of the
ordered pair and X 2 = the value of the second component of the ordered pair. Make use of the linearity
property from Equation (4).
95. At a gambling casino, a ball will be drawn from a bin containing 43 red balls, 27 green balls, and 8 blue
balls. A player marks a game card with the color he or she believes will be picked. The prize money for
guessing the correct color is
Red
$3.00
Green
$6.00
Blue
$10.00
The price of the game card is $5.00. Find the expected value of the prize money.
96. A directory on a computer’s hard disk contains 12 files, 3 of which have viruses. If a file with a virus is
selected, the virus is detected and another file is then selected. Find the expected number of files that must
be selected in order to get a virus-free file.
97. Bit strings are sent across a computer network in packets of length 10. The probability of a bit getting corrupted (that is, a 0 gets changed to a 1 or vice versa) is 0.01. These bit errors are independent.
a. Find the probability that in a single packet there are no bit errors. (Hint: It’s OK to call a bit error a
“success.”)
b. Find the probability that there are no more than two bit errors.
c. Find the probability that there is at least one bit error.
98. Of the items produced in a certain manufacturing facility, 5% are defective. If 8 items are chosen at random, find the probability that
a. 1 is defective.
b. 2 are defective.
c. none is defective.
d. at least 1 is defective.
e. at most 1 is defective.
99. A student has failed to study for a true−false test and guesses at every one of the 10 questions. If the passing grade is 8 correct answers, what is the probability that the student will pass the quiz?
100. A baseball player has a probability p = 0.04 of hitting a home run for each at bat. Find the minimum
number of at bats he must take so that there is at least an 80% probability of hitting a home run (that is, at
least 1 home run).
101. Find the average number of comparisons to search for a target element x using the sequential search algorithm under the assumption that x is equally likely to be at any of the n positions in the list or not in the
list.
102. Find the average number of comparisons to search for a target element x using the sequential search algorithm under the assumption that x is not in the list 80% of the time, but if x is in the list it is equally likely
to be at any of the n positions.