Download 1. Test question here

Mu Alpha Theta National Convention: Hawaii, 2005
Probability Topic Test – Alpha Division
1.
How many distinguishable permutations of the letters of the word “Hawaii” are there?
A) 720
2.
B) 360
C) 24
D) 96
E) NOTA
Three dice are rolled. What is the probability that they all show the same number?
A)
1
216
B)
1
36
C)
1
6
D)
1
72
E) NOTA
Questions 3-5 refer to a multiple-choice test where each question has 5 possible answers, exactly
one of which is correct (similar to this test).
3.
If a student guesses a random answer on a single question, what is the probability that the
student guesses correctly on that question?
A)
4.
1
4
C)
1
25
D)
1
5
E) NOTA
3
13
B)
1
4
C)
1
20
D)
1
5
E) NOTA
Now suppose that A-D are equally likely, but each of those is thrice as likely as E. If the
student limits his guess to A-D, what is the probability that the student guesses correctly on
a single question?
A)
6.
B)
If the answers are randomly distributed, but the student limits his guess to A-D, what is the
probability that the student guesses correctly on a single question?
A)
5.
1
2
1
5
B)
1
4
C)
3
13
D)
1
20
The probability that Isaac goes grocery shopping on Thursday is
E) NOTA
1
. The probability that
10
1
. Assuming independence, what is the
5
probability that both of them go grocery shopping on Thursday?
Rebekah goes grocery shopping on Thursday is
A)
1
2
B)
1
10
C)
3
10
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D)
1
50
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Probability Topic Test – Alpha Division
7.
An experiment consists of flipping three distinguishable coins and observing whether each
coin is heads or tails. How many elements are in the sample space of this experiment?
A) 3
8.
C) 4
D) 7
E) NOTA
A random subset of the 9 smallest natural numbers is chosen. What is the probability that
the chosen subset contains the number 5?
A)
9.
B) 2
1
19
B)
1
2
C)
1
9
D)
1
32
E) NOTA
Two dice are rolled and at least one die is observed to be a 2. What is the probability that
the sum of the dice is 6?
A)
1
6
B)
5
36
C)
2
11
D)
5
216
E) NOTA
10. Two integers, R and S (not necessarily distinct), are chosen at random between 1 and 100
R
inclusively. What is P(  1 )?
S
A)
1
2
B)
49
100
C)
99
200
D)
101
200
E) NOTA
11. What is the probability that a random 3-digit (no leading 0’s) positive integer is divisible by
8?
A)
1
8
B)
113
900
C)
28
225
D)
19
150
E) NOTA
12. A sequence of numbers is generated as follows: the first number is 0; for each subsequent
number, a die is rolled and that outcome is added to the previous number in the sequence.
What is the probability that the sequence contains the number 4?
A)
1
6
B)
343
1296
C)
307
1296
D)
1
4
E) NOTA
13. If P(E)=.15, P(F)=.09 and P(E  F)=.19, find P(EC  F).
A) .07
B) .04
C) .24
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D) .02
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Probability Topic Test – Alpha Division
14. A random subset of the ten smallest natural numbers is chosen. What is the probability that
the largest element of the chosen subset is even (the empty set fails vacuously)?
A)
1
2
B)
341
512
C)
31
32
D)
511
1024
E) NOTA
For questions 15-17, urn I contains 2 red balls and 3 white balls while urn II contains 7 red balls
and 3 white balls. An experiment consists of (somehow) choosing an urn and then randomly
selecting a ball from that urn.
15. If the urn is chosen at random, what is P(ball is red)?
A)
11
20
B)
3
5
C)
7
25
D)
7
9
E) NOTA
16. With what probability must urn I be chosen to ensure P(urn I was chosen | ball is red) = .5?
A)
1
5
B)
7
10
C)
7
11
D)
3
5
E) NOTA
17. Two balls are chosen at random from urn II (without replacement). What is the probability
that they are the same color?
A)
7
15
B)
1
10
C)
2
3
D)
8
15
E) NOTA
18. How many 10-digit (no leading 0’s) positive binary integers have an even number of 1’s?
A) 512
B) 511
C) 255
D) 256
E) NOTA
19. A six-sided die and a four-sided die (numbered 1-4) are rolled and their sum is taken. What
is the expected value of this sum?
A) 6
20. If P ( F ) 
A)
3
125
B) 10
C) 2
D) 7
E) NOTA
1
4
3
1
, P (G )  , P ( E  F  G ) 
and P( E C  F  G )  , what is P( F | G ) ?
2
5
10
10
B)
1
2
C)
2
5
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D)
1
3
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Probability Topic Test – Alpha Division
21. Four friends meet for lunch and each friend puts his car key in the middle of their table.
After lunch, each friend grabs a key at random. What is the probability that at least one
person gets his own key?
A)
5
8
B)
1
4
C)
1
24
D)
3
8
E) NOTA
22. In how many ways may 6 distinguishable crayons be placed into 5 distinguishable boxes if
no box may be left empty?
A) 15625
B) 7776
C) 3600
D) 1800
23. The diagram to the right shows all of the streets in Fictitiousville, ID
(four roads running N to S and four roads running E to W). If a person
starts at point A (the NW corner of the town) and walks to point B (the
SE corner of the town), traveling ONLY along the streets and ONLY in
the south or east directions, what is the probability that the person will not
travel through point C (one block SW of the NE corner of town)?
A)
1
4
B)
11
20
C)
1
5
D)
9
20
E) NOTA
A
C
B
E) NOTA
24. Two distinct integers are chosen at random from the interval [1, 7]. What is the probability
that their sum exceeds 10?
A)
4
21
B)
2
21
C)
1
7
D)
2
7
E) NOTA
25. A bag contains three red marbles, four white marbles, and five blue marbles. When three
marbles are chosen at random, what is the probability that they are all the same color?
A)
1
22
B)
3
44
C)
4
55
D)
7
110
E) NOTA
26. Four distinguishable one-sided keys (a one sided key has one edge that is
“flat” and one edge that is “bumpy”) are placed on a key ring. In how
many distinguishable ways may this be done?
A) 24
B) 48
C) 96
Page 4 of 5
D) 384
E) NOTA
Mu Alpha Theta National Convention: Hawaii, 2005
Probability Topic Test – Alpha Division
27. What is the probability that a random five-card poker hand contains the Deuce of Spades?
A)
1
5
B)
5
52
C)
1
2
D)
1
2598960
E) NOTA
28. What is the probability that a random permutation of the letters of the word HAWAII has the
consonants as the first two letters?
A)
1
360
B)
1
180
C)
1
3
D)
1
5
E) NOTA
29. Dick and Jane agree to meet at a restaurant for lunch. Each of them will arrive and a random
time between 12:00PM and 1:00PM. Dick is willing to wait 10 minutes for Jane to arrive
and Jane is willing to wait 20 minutes for Dick to arrive. What is the probability that they
are both at the restaurant at the same time?
A)
5
12
B)
1
2
C)
31
72
D)
4
9
E) NOTA
30. What is the probability that a randomly chosen positive integer in the interval
[10000, 99999] contains only odd digits?
A)
1
2
B)
1
32
C)
5
18
Page 5 of 5
D)
5
144
E) NOTA