Download For Questions 1-3 use the following information. Set S

Open Probability
State 2015
Specific instructions: Pick the exact answer unless otherwise stated. NOTA denotes none of the above.
For Questions 1-3 use the following information. Set S consists of the first 100 positive
integers.
1) What is the probability of picking a number that is both prime and odd from set S?
a) .17
b) .24
c) .25
d) .27
e) NOTA
2) What is the probability of picking a triangular number that is odd from set S?
a) .07
b) .13
c) .24
d) .25
e) NOTA
3) What is the probability of choosing a Fibonacci number from set S? Given: 11th
Fibonacci number is 89
a) .08
b) .09
c) .10
d) .11
e) NOTA
4) Twelve twins are participating in a tournament. The 24 people are randomly assigned
to groups of 3. What is the expected number of twins in the same group?
a) 23/24
b) 24/23
c) 1
d) 3/2
e) NOTA
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5) Given 𝑃(𝐴) = 2, 𝑃(𝐵) = 2, and 𝑃(𝐵|𝐴) = 3. What is 𝑃(𝐴 𝑎𝑛𝑑 𝐵)?
a) 1/6
b) 1/3
c) 1/2
d) 2/3
e) NOTA
6) Let X equal to the number of ways that a committee of 49 people can be chosen out of
99 people and Y equal to the number of ways a different committee of 49 can be formed
out of the same 99 people. What is X+Y? Note: Being chosen to one committee doesn’t
disqualify you for the other committee
a) 99!/2!
99
b) (49
)
100
c) ( 50 )
99 2
d) ((49
))
e) NOTA
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Open Probability
State 2015
7) A bag contains four balls of the following colors: Green, Yellow and two Blue. If two
balls were drawn at random and one is known to be Blue, what is the probability the
other is Blue?
a) 1/6
b) 1/5
c) 1/2
d) 5/6
e) NOTA
8) A dot starts its life on the bottom right corner of an 8x8 grid. How many different ways
can it move to the top left most corners in the least number of moves, if the dot can only
go up/down or left/right?
a) 14
b) 128
c) 3432
d) 6864
e) NOTA
9) If A, B, and C are positive integers, how many solutions are there for the following
equation?
𝐴 + 𝐵 + 𝐶 = 20
a) (19
)
2
b) (23
)
3
c) 20 ∗ 19 ∗ 18
d) (20
)
3
e) NOTA
10)A math student is about to play the lottery: The ticket is a combination of a single letter
(A-Z) and three numbers (0-9). The letter is first then the three numbers.
The three ways to win the lottery:
Match the last two characters on the ticket: Win $100
Match the last three characters on the ticket: Win $1000
Match all the characters on the ticker: Win $100,000
What is the expected payout if the ticket costs $20?
a) -18
b) -184/13
c) -368000
d) 0
e) NOTA
11)What is the probability of choosing an even number from the first ten rows of the
Pascal’s Triangle? Row 1 is 1 and Row 2 is 1,1
a) 2/5
b) 3/5
c) 4/11
d) 7/11
e) NOTA
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Open Probability
State 2015
12)A fair six-sided die is rolled 3 times. What is the probability of NOT rolling a number
larger than 4 on any of the rolls?
a) 1/2
b) 1/4
c) 1/8
d) 1/16
e) NOTA
13)What is the probability of drawing a Queen and then a Hearts after replacement from a
standard deck of 52 cards? The cards are randomized so that every card has the same
probability of being chosen.
a) 1/52
b) 1/68
c) 1/48
d) 1/72
e) NOTA
14)How many unique 5 digit palindrome numbers are there? Note: 00001 will be
considered a 5 digit number
a) 1
b) 100
c) 900
d) 1000
e) NOTA
15)When a pair of fair 6-sided die are rolled, what is the probability of rolling a two even
numbers?
a) 1/2
b) 1/4
c) 1/6
d) 1/8
e) NOTA
16)What is the sum of the coefficients in the expanded form of (4𝑥 + 𝑦)5
a) 64
b) 1024
c) 3125
d) 7776
e) NOTA
17)Diego and Ryan are having a contest to see who can toss the most heads on a fair coin.
Diego only wins when he has more heads than Ryan, otherwise Ryan wins. Diego likes
to cheat so he gets an extra toss than Ryan. What is the probability that Ryan wins?
Note: Ryan muss toss at least once
a) 1/5
b) 1/3
c) 1/2
d) 0
e) NOTA
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Open Probability
State 2015
18)A box contains 10 white chocolates and 15 dark chocolates. If each chocolate is
removed one by one and eaten. What is the probability of the last one being a dark
chocolate?
a) 1/2
b) 1/3
c) 1/4
d) 1/5
e) NOTA
19)Diego lives a lonely life on a one-dimensional number line. He can only move forward
two or back one. If he starts at x=0 and ends up at x=7, how many distinct ways could he
have taken if he took at most 8 steps? Note: Forward two counts as a step and backward
one counts as a step
a) 5
b) 40
c) 56
d) 61
e) NOTA
20)6 dots are placed at random on a 10x10 grid. What is the number of combinations that
no two rows or columns contain more than one dot? Note: Only one dot can be in any
one position
a) 100!
b) (10
)
6
100
c) ( 6 )
d) ((10
))2
6
e) NOTA
21)A string is cut at two points chosen randomly on the string. What is the probability that
no piece is more than half the length of the original string?
a) 1/2
b) 1/4
c) 2/3
d) 3/5
e) NOTA
22)Ryan stole his friend’s iPhone. He’s trying to figure out the passcode combination to
unlock the phone. The passcode is a four digit number, with each digit only being a
number from 0 to 9 (inclusive). If he knows that each digit has to be at least one away
and 0 and 9 cannot be next to each other (1234, 0909, 1272, and 1278 cannot be
combinations), how many possibilities are there for the passcode?
a) 512
b) 732
c) 1024
d) 5120
e) NOTA
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Open Probability
State 2015
23)A fair coin is tossed 20 times. The probability of getting three or more heads in a row is
0.7870 and the probability of getting (three or more heads in a row) or (three or more
tails in a row) is 0.9791. What is the probability of getting three or more heads in a row
and three or more tails in a row?
a) 0.1609
b) 0.5949
c) 0.7870
d) 0.9791
e) NOTA
24)This morning when a MAO student was getting ready for this test he decided to count
the number of outfits he has. If he has 3 math team shirts, 10 pants and 5 pairs of shoes.
How many combinations of outfits are there for him to wear to the competition? Note:
His sponsor is Mrs. Woolfe so he has to wear a math team shirt and no two clothing
items are identical.
a) 30
b) 50
c) 150
d) 300
e) NOTA
25)What is the probability of getting the word “Rearrangement” from arranging 2 A’s, 3 E’s,
1 G, 1 M, 2 N’s, 3 R’s and 1 T?
1
a) 13!
b)
c)
1
2!3!2!3!
13!
2!3!2!3!
2!3!2!3!
d) 13!
e) NOTA
26) In the state of Washington, the House of Representatives has 10 members. If 8
republicans and 6 democrats are running and the House must consist of 6 Republicans
and 4 Democrats, how many combinations of Representatives can be elected?
a) (86)
b) (64)
c) (86)(64)
d) (14
)
10
e) NOTA
27) A family has 3 children. What is the probability of the three children being a boy and
two girls if one of the three children if at least one of the three is a boy?
a) 3/8
b) 3/7
c) 1/5
d) 7/8
e) NOTA
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Open Probability
State 2015
For the next question use the following information: Assume a class of N students and K
seats. Each student gets an assigned seat (meaning N is less that or equal to K). If the
first student to enter the class chooses a random seat to sit in and every student after
either sits in the assigned seat if unoccupied or random seat if occupied.
28)What is the probability of the last person to enter the classroom will not sit in the
assigned seat if there are 25 students and 25 seats? Pick the answer that is closest to the
actual value.
a) .75
b) .5
c) .25
d) 0
e) NOTA
29)In a poker game there are four players: A, B, C, and D. The dealer is chosen by following
these rules: In the following order (ABCDABCDABCD…), each player is dealt a card and
the first player to receive the ace of spades or the ace of hearts from a well shuffled
standard deck of 52 cards. Who has the highest probability of being the dealer?
a) Person A
b) Person B
c) Person C
d) Person D
e) NOTA
30)Whose name is on the visual representation of the binomial theorem?
a) Blaise Pascal
b) Pierre de Fermat
c) Girolamo Gardano
d) Chevalier de Mere
e) NOTA
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