Stats 2D03 Practice problems (1)
1. A person is holding 7 different playing cards. They are to keep 4 cards in their hand, throw 2 cards
away, and place one card face up in the middle of the table. How many ways can this be done?
2. An urn contains 3 yellow and 5 red balls. Two balls are withdraw from the urn without replacement.
What is the probability that the two balls drawn have different colours?
3. Suppose that you are in a restaurant, and are going to have either soup or salad but not both. There
are 2 soups and 4 salads on the menu. how many choices you will have?
4. How many words of length 5 can you form, from an alphabet letters?
5. How many words of length 5 can you form, from an alphabet letters without replacement?
6. Erin has 5 tops, 6 skirts and 4 caps from which to choose an outfit. In how many ways can she select
one top, one skirt and one cap?
7. In how many ways can 5 boys and 4 girls be arranged on a bench if
(a) There are no restrictions?
(b) Boys and girls alternate?
(c) Boys and girls are in separate groups?
8. How many outcomes you get if you flip a fair coin 10 times?
9. In how many ways can 3 different history books, 5 different math books, and 6 different novels be
arranged on a shelf if the books of each type must be together?
10. How many different 7-person committees can be made if there are 150 students and 50 faculty members,
if the committee must contain more students than faculty?
11. 50 applications for a job have been received by a hiring manager, and each has to be reviewed by one
member of the department. In order to share the work, the manager gets Al, Beth, Chris and Dawn
to each take 9 of the applications to be reviewed; the manager takes the rest. How many ways can the
different applications divided in this manner?
12. A die is rolled 4 times. What is the probability that the number 5 comes up exactly two times?
13. Final exams will be given over a period of 8 days, with 3 slots per day: morning, afternoon and
evening. Exams of different classes cannot be written in the same slot. Consider a student that has
five classes, each with a final exam. How many different final exams schedules are possible? (note that
first ”calculus”, then ”probability” is different from first ”probability” then ”calculus”).
14. Each of the 8 letters in the word PROBABLE is written on a separate tile. How many different 3 letter
”words” (they don’t have to be real words) can be made from 3 of those tiles if a word must contain
at least one vowel?
15. An urn has 4 blue, 2 yellow and 3 red balls. A ball is selected at random, the colour is noted, and the
ball is returned to the urn. If this is done four times, what is the probability that a red ball was drawn
at least twice?
16. There are 6 red and 4 blue balls inside an urn. Three balls are selected at random without replacement.
Find the probability of the event that more red balls are selected.
17. What is the probability of P(A ∪ B ∪ C) if the following probabilities are known
P(A) = 30%, P(B c ) = 60%, P(C) = 25%,
P(A ∩ B) = 10%, P(A ∩ C) = 8%,
P((B ∩ C) ∪ A) = 31%, P((B ∩ C ∩ A)c ) = 95%
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18. List all possible arrangements of the four letters m, a, r, and y. Let C1 be the collection of the
arrangements in which y is in the last position. Let C2 be the collection of the arrangements in which
m is in the first position. Find the union and intersection of C1 and C2 .
19. A bowl contains 16 chips, of which 6 are red, 7 are white, and 3 are blue. If four chips are taken at
random and without replacement, find the probability that:
(a) Each of the four chips is red.
(b) None of four chips is red.
(c) There is at least one chip of each color.
20. (a) A couple has two children. What is the probability they are both girls if you know that the older
of the two is a girl?
(b) A different couple has two children. What is the probability they are both girls if you are given
that at least one of them is a girl?
21. A Euchre deck has 24 cards, consisting of the 9, 10, Jack, Queen, King and Ace of the usual four suits.
If a hand of 5 cards is dealt, what is the probability that it contains two pairs?
22. An urn contains 11 red balls numbered 1 through 11, 11 yellow balls numbered 1 through 11, 11 green
balls numbered 1 through 11, and 11 black balls numbered 1 through 11. If 4 balls are randomly
selected, find the probability of getting
(a) a flush (i.e., all balls the same color).
(b) three of a kind. (Three of a kind is 3 balls of one denomination and a fourth ball of a different
denomination. e.g., 5,5,5,2)
(c) two pairs. (A pair is two balls of the same denomination. e.g., 3,3. Two pairs is something like
3,3,5,5, i.e., a pair of one denomination and a second pair of a different denomination.)
23. Suppose that 60% of police calls concern domestic disputes. Suppose further that 92% of police calls
do not result in an arrest. If only 3% of domestic disputes result in an arrest, what is the probability
that a police call that does not concern a domestic dispute does not result in an arrest?
24. A random experiment consists of drawing a card from an ordinary deck of 52 playing cards. Let the
probability set function P assign a probability of 1/52 to each of the 52 possible outcomes. Let A1
denote the collection of the 13 hearts and let A2 denote the collection of the 4 kings.
(a) Compute P(A1 ).
(b) Compute P(A2 ).
(c) Compute P(A1 ∩ A2 ).
(d) Compute P(A1 ∪ A2 ).
25. Suppose the test for HIV is 99% accurate in both directions and 0.3% of the population is HIV positive.
(a) If we select a random person to get tested for HIV, what is the overall probability that they test
positive?
(b) If someone tests positive, what is the probability they actually have HIV?
26. A population of people can be divided into three groups: the 0.3% who currently have the disease
Statistigitis, the 2.0% who had Statistigitis but who are now recovered, and the other 97.7% who have
never yet had Statistigitis. A test for the disease has been created for which people currently with the
disease test positive 90% of the time, while those who have never had it test negative 99.9% of the
time. People who had the disease but are now recovered test positive 25 % of the time. If someone
tests positive for Statistigitis, what is the probability that they currently have the disease?
27. There are three coins sitting on a table. Two are fair and one has Heads on both sides. One coin is
selected at random and is flipped four times, getting Heads each time. What is the probability the
coin selected is the two-headed coin?
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