Statistics: Probability
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1. A bag has blue, green, and yellow marbles. There are 3 times as many green marbles as yellow marbles and 4 times as many blue marbles as green marbles. What is the probability of randomly selecting a yellow marble?

(A) 1 / 16
(B) 1 / 15
(C) 1 / 12
(D) 1 / 8
(E) 1 / 7

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Let x be the number of yellow marbles. The bag has 3x green and 12x blue and 16x total. The probability of getting yellow is x/(16x) or 1/16.

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2. A bag contains N marbles. A marble selected at random from the bag could be one of m different colors. There are an equal number of marbles of each color in the bag. What is the minimum number of marbles you must select at random from the bag in order to be 100% certain you will have (outside the bag) a pair of marbles of the same color?

(A) N / m + 1
(B) m + 1
(C) N / x + 1
(D) x + 1
(E) N / (mx) + 1

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Suppose you have 6 or each color and suppose there are three colors (that's m). The worst case scenario is you pick each of the three colors on your first three picks. Finally after your fourth pick, you are guaranteed to have a pair. So the answer is m+1 or (B).

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3. Another blue, green, and yellow marble bag. The probability of picking a blue marble is 1/2. The probability of picking a green marble is 1/3. There are 20 yellow marbles in the bag. Which of the following is true about the bag of marbles?

I. There are 40 blue marbles.
II. There are 30 green marbles.
III. The probability of picking a yellow marble is 1/5.

(A) none
(B) I, only
(C) II, only
(D) III, only
(E) I, II, and III

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The total probability is always 1 so your chances of picking a yellow marble is 1/6. That means there are 120 marbles in the bag, 60 of which are blue and 40 of which are green. So none of the statements are true. The answer is (A).

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4. Yet another bag, same colors. There are twice as many green marbles as yellow marbles. The probability of picking a blue marble is 4/5. The number of green marbles is x. What is the number of blue marbles?

(A) 15x
(B) 12x
(C) 7.5x
(D) 6x
(E) 4x

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Since blue is 4/5, the probability of picking yellow or green is 1/5. This 1/5 has to be divided into 3 parts since there's 2 parts green compared to 1 part yellow. So green is 2/15 and yellow is 1/15 (total is still 3/15 = 1/5). Blue has to be 12/15 which is 6 times the probability of picking green, so there must be 6x blue marbles. The answer is (D).

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5. The evil blue, green, and yellow marble bag has an equal number of blue and green marbles. There are four times as many yellow marbles as blue marbles. Some of the marbles are solid and some are hollow. For each color, there are five times as many solid marbles as hollow marbles. What is the probability of randomly choosing a solid green marble?

(A) 7 / 20
(B) 5 / 36
(C) 4 / 27
(D) 4 / 25
(E) 2 / 15

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Let's there are x blue marbles, x green marbles, 4x yellow marbles, and 6x total marbles. Since there are 5 times as many solids as hollows that means solids are 5/6 and hollows are 1/6. Since there are x green marbles, there are (5/6)x green marbles out of 6x total marbles. Dividing you get the probability of getting solid green is 5/36 or (B).

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6. A slightly more interesting bag with blue, green, and yellow marbles has some marbles that explode when removed from the bag. Only blue marbles explode. There are 162 blue marbles in the bag, not all of which explode. Two-thirds of the marbles in the bag are blue. If one marble is chosen at random from the full bag, the probability that it will explode is 1/9. What is the ratio of exploding blue marbles to all blue marbles?

(A) 2 / 27
(B) 2 / 9
(C) 1 / 9
(D) 1 / 6
(E) 1 / 3

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Since 162 marbles is 2/3 of the bag, there are 162+81=243 marbles in the bag and 1/9 of these explode. So 27 marbles explode and all of these are blue. The ratio they are asking for is 27/162 = 1/6. The answer is (D).

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7. If the probability of picking a yellow marble out of a bag is 1/n and n is an integer greater than 1, what is the ratio of the number of yellow marbles in the bag to the number of other marbles?

(A) 1 / n
(B) n
(C) n / (n–1)
(D) 1 / (n–1)
(E) 1 / (n+1)

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Suppose n is 5. That means 1 out of 5 marbles is yellow and 4 out of 5 are some other color. So the ratio of yellow to other is 1 to 4 which is 1 to (n–1).

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8. All of the numbers from 1 to 100 inclusive are written in a single line. If you choose a digit at random from the line of digits, what is the probability that you would choose a 1?

(A) 1 / 8
(B) 3 / 20
(C) 4 / 35
(D) 5 / 48
(E) 7 / 64

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The important numbers are: 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, and 100. There are twenty-one 1's. Between 1 and 100 there are nine 1-digit numbers, one 3-digit number, and ninety 2-digit numbers (one hundred numbers all together). So there are 9 + 3 + 180 = 192 digits. The probability of picking a 1 is 21/192 which reduces to 7/64. The answer is (E).

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9. A randomly-selected group of people were asked the following question: "Do you think the average high school principal could run a successful business?" The results were as follows: 240 people said NO; 100 people said MAYBE; 20 people said YES. Based on this survey, if you asked another 90 randomly-selected people the same question, how many would be expected to say NO?

(A) 45
(B) 60
(C) 72
(D) 75
(E) 80

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The fraction that said NO is 240/360 = 2/3. Out of 90 you would expect 2/3 or 60 to say NO. So the answer is (B).

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10. According to one study, the probability that an American adult ate too much sugar as a child is 80%. According to a companion study, 30% of people who ate too much sugar as children have trouble controlling their weight as adults while only 10% of people who did not eat too much sugar as children have a weight control problem. According to both studies, how many American adults out of 100 would be expected to have trouble contolling their weight?

(A) 20
(B) 22
(C) 24
(D) 26
(E) 28

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Assume you have 100 children: 80 ate too much sugar while 20 did not. Of the 80 that ate too much sugar, 30% or 24, will have trouble with their weight. Of the 20 who ate healthy as kids, 10% or 2 will have weight troubles. That's 26 out of 100 or (D).

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