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IE241 Problems
1. Three fair coins are tossed in the air.
What is the probability of getting 3
heads when they land?
2. A town has 5000 adults and a random
sample of 100 are asked for their
opinion about a proposed sports
arena. 60 of the respondents voted
for it and 40 against it. If in fact the
townspeople were half in favor and
half against, what is probability that
the sample would show 60% in favor?
3. If a box contains 40 good fuses and
10 defective fuses, and 10 fuses are
chosen from the box at random. What
is the probability that all 10 are good?
4. A die has two of its sides painted red,
two painted yellow, and two painted
black? If the die is rolled twice, what
is the probability of two reds?
5. Assume that the ratio of male
children is ½. In a family where 6
children are desired, what is the
probability (a) that all 6 children will be
of the same sex, and (b) that exactly 3
will be boys and 3 girls?
6. Compare the chances of rolling a 4 with 1
die and rolling a total of 8 with 2 dice.
7. The probability = ½ that a finesse in
bridge will be successful. What is the
probability that 3 out of 5 finesses will
be successful?
8. A card is drawn from an ordinary deck.
(a) What is the probability that it is a
king, given that it is a face card
(K,Q,J)?
(b) What is the probability that it is a
black king, given that it is a face card?
9. Two balls are drawn from an urn
containing 2 white balls, 3 black balls,
and 4 green balls. What is the probability
that the first is white and second is black?
10. Consider a deck of cards consisting of only
A, K, Q, J of each of the four suits. Find (a)
the marginal distribution of suit and (b) the
marginal distribution of face cards.
11. A company manufactures transistors in
three different plants A, B, C, which use very
similar procedures. It is decided to inspect
the transistors in plant A because it is the
largest. In order to test a week’s production,
100 transistors are selected at random and
tested for defects. If 2 defects are found,
what exactly can the company conclude
about its transistor manufacturing operation?
12. A test of the breaking strength in
pounds of 3/16” manila rope in a
sample of 100 ropes produced an
estimated mean of 550 and an
estimated standard deviation of 80.
Find a 95% confidence interval for the
mean, assuming that the distribution of
breaking strength is normal.
13. Compare the average length of a 95%
confidence interval for the mean of a
normal population based on the t
distribution with the length of the
confidence interval if σ were known.
14. Show that the length of the
confidence interval based on t will
approach 0 as n increases without
bound.
15. A professor believes in the principle
of a sound mind in a sound body, and
decides to prove his point by measuring
male students in his class on their
running speed.
He can’t measure the entire class so
he decides to take the first 10 male
students who show up for class on
Monday. If he does this, how can he
generalize his conclusions?
16. The diameters in feet of 56 shrubs were
measured with the following results, where X
= diameter and freq(X) = the frequency with
which the value x occurred.
X
1
2
3
4
5
6
7
8
9
12
freq (x) 1
7
11 16
8
4
5
2
1
1
Draw the histogram and calculate the mean
diameter, the median diameter, and the mode.
17. Draw the histogram for the sum of
the face numbers that come up when
rolling two honest dice. What is the
mean, median, and mode of this
distribution?
18. Given that a binomial distribution
has mean = 12 and variance = 8, find
its parameters.
19. If one point on a binomial cdf is
(3, 0.17), what is the probability of a
value ≥ 4 ?
20. If one point on a normal cdf =
(1.2, 0.37), what is the probability of a
value ≥ 1.2 ?
21. Suppose you have two sets of data
for the random variable X. The first set
consisted of n1 observations and had
mean X1 and standard deviation s1 and
the second set consisted of n2
observations with mean X 2 and standard
deviation s2. What is the mean of the
combined group?
22. Suppose the November weather
records show on average 3 out of 30
days of rain. Consider each day of
November as an independent trial and
compute the probability of at most 2
rainy days next November.
23. In the baseball world series, a team
must win 4 games to win the series.
The odds are 2:1 that team A will win
each game. (a) What is the probability
that team A wins in 4 straight games?
(b) What is the probability that the
series ends in 4 games?
(c) What is probability that team A wins
the series in 7 games?
24. A man takes 20 shots at a target.
His probability of hitting the target is
1/10. What is the probability of at least
2 hits?
25. Give an example of two random
variables for which the variance of their
sum is (a) larger than the sum of their
variances and (b) smaller than the sum
of their variances.
26. Past experience indicates that wire
rods purchased from company A have a
mean breaking strength of 400 pounds
and a standard deviation of 15 pounds.
If 16 rods are selected, between what
two values could you reasonably expect
their mean to be?
27. An urn contains 2 red balls, 3 green
balls, and 4 black balls. If 4 balls are
drawn with replacement, what is the
probability of getting 1 red, 1 green,
and 2 black balls?
28. Three dice are thrown simultaneously.
What is the probability of getting a total
of 6 on the three dice?
29. Explain why it would not be
surprising to find a correlation between
traffic on Wall Street and tide height in
Maine if you observed every hour from
6 am to 10 pm and high tide occurred
at 8 am.