Some fun probability problems

[Oldman]

These are for Mick, since he's at the 800 level now. Hope these help bro! I'll post the answers after folks have had a chance to work them out. These are from a document on a GMAT forum. I thought they would be fun to mess with. I have about 20 more if you guys like these. They don't seem particularly difficult and seem like the level one could expect on the GRE.

1. In a workshop there are 4 kinds of beds, 3 kinds of closets, 2 kinds of shelves and 7 kinds of chairs. In how many ways can a person decorate his room if he wants to buy in the workshop one shelf, one bed and one of the following: a chair or a closet?

a) 168.
b) 16.
c) 80.
d) 48.
e) 56.

2. In a workshop there are 4 kinds of beds, 3 kinds of closets, 2 kinds of shelves and 7 kinds of chairs. In how many ways can a person decorate his room if he wants to buy in the workshop one shelf, one bed and one of the following: a chair or a closet?

a) 168.
b) 16.
c) 80.
d) 48.
e) 56.

3. Three people are to be seated on a bench. How many different sitting arrangements are possible if Erik must sit next to Joe?

a) 2.
b) 4.
c) 6.
d) 8.
e) 10.

4. How many 3-digit numbers satisfy the following conditions: The first digit is different from zero and the other digits are all different from each other?

a) 648.
b) 504.
c) 576.
d) 810.
e) 672.


5. Barbara has 8 shirts and 9 pants. How many clothing combinations does Barbara have, if she doesn’t wear 2 specific shirts with 3 specific pants?

a) 41.
b) 66.
c) 36.
d) 70.
e) 56.
6. A credit card number has 6 digits (between 1 to 9). The first two digits are 12 in that order, the third digit is bigger than 6, the forth is divisible by 3 and the fifth digit is 3 times the sixth. How many different credit card numbers exist?

a) 27.
b) 36.
c) 72.
d) 112.
e) 422.

7. In jar A there are 3 white balls and 2 green ones, in jar B there is one white ball and three green ones. A jar is randomly picked, what is the probability of picking up a white ball out of jar A?

a) 2/5.
b) 3/5.
c) 3/10.
d) 3/4
e) 2/3.

8. Out of a box that contains 4 black and 6 white mice, three are randomly chosen. What is the probability that all three will be black?

a) 8/125.
b) 1/30.
c) 2/5.
d) 1/720.
e) 3/10.

9. The probability of pulling a black ball out of a glass jar is 1/X. The probability of pulling a black ball out of a glass jar and breaking the jar is 1/Y. What is the probability of breaking the jar?

a) 1/(XY).
b) X/Y.
c) Y/X.
d) 1/(X+Y).
e) 1/(X-Y).

10. Danny, Doris and Dolly flipped a coin 5 times and each time the coin landed on “heads”. Dolly bet that on the sixth time the coin will land on “tails”, what is the probability that she’s right?

a) 1.
b) ½.
c) ¾.
d) ¼.
e) 1/3.

11. In a deck of cards there are 52 cards numbered from 1 to 13. There are 4 cards of each number in the deck. If you insert 12 more cards with the number 10 on them and you shuffle the deck really good, what is the probability to pull out a card with a number 10 on it?

a) 1/4.
b) 4/17.
c) 5/29.
d) 4/13.
e) 1/3.

12. There are 18 balls in a jar. You take out 3 blue balls without putting them back inside, and now the probability of pulling out a blue ball is 1/5. How many blue balls were there in the beginning?

a) 9.
b) 8.
c) 7.
d) 12.
e) 6.

13. In a box there are A green balls, 3A + 6 red balls and 2 yellow ones.
If there are no other colors, what is the probability of taking out a green or a yellow ball?

a) 1/5.
b) 1/2.
c) 1/3.
d) 1/4.
e) 2/3.


14. The probability of Sam passing the exam is 1/4. The probability of Sam passing the exam and Michael passing the driving test is 1/6.
What is the probability of Michael passing his driving test?

a) 1/24.
b) 1/2.
c) 1/3.
d) 2/3.
e) 2/5

15. In a blue jar there are red, white and green balls. The probability of drawing a red ball is 1/5. The probability of drawing a red ball, returning it, and then drawing a white ball is 1/10. What is the probability of drawing a white ball?

a) 1/5.
b) ½.
c) 1/3.
d) 3/10.
e) ¼.

16. Out of a classroom of 6 boys and 4 girls the teacher picks a president for the student board, a vice president and a secretary. What is the probability that only girls will be elected?

a) 8/125.
b) 2/5.
c) 1/30.
d) 1/720.
e) 13/48.

17. Two dice are rolled. What is the probability the sum will be greater than 10?

a) 1/9.
b) 1/12.
c) 5/36.
d) 1/6.
e) 1/5.

18. The probability of having a girl is identical to the probability of having a boy. In a family with three children, what is the probability that all the children are of the same gender?

a) 1/8.
b) 1/6.
c) 1/3.
d) 1/5.
e) ¼.

19. On one side of a coin there is the number 0 and on the other side the number 1. What is the probability that the sum of three coin tosses will be 2?

a) 1/8.
b) ½.
c) 1/5.
d) 3/8.
e) 1/3.

20. In a flower shop, there are 5 different types of flowers. Two of the flowers are blue, two are red and one is yellow. In how many different combinations of different colors can a 3-flower garland be made?

a) 4.
b) 20.
c) 3.
d) 5.
e) 6.

[mickgreen58]

Thanks Oldman, I will do the ones that I don't know.

[mickgreen58]

#5 - Not sure how to approach this problem.

#6 - not sure why the answer isn't 3^4. I would think it would be 1x1x3x3x3x3.

#7 - B

#8 - not sure why it isn't 4/10 x 3/9 x 2/8?

#9 - I would think (1/x*1/y) / 1/y = 1/y?

[Oldman]

Here are the OAs:

Explanations:

1. The best answer is C.
You must multiply your options to every item. (2 shelves) x (4 beds) x (3 closets + 7 chairs) = 80 possibilities.

2. The best answer is C.
You must multiply your options to every item. (2 shelves) x (4 beds) x (3 closets + 7 chairs) = 80 possibilities.


3. The best answer is B.
Treat the two who must sit together as one person. You have two possible sitting arrangements. Then remember that the two that sit together can switch places. So you have two times two arrangements and a total of four.

4. The best answer is C.
For the first digit you have 9 options (from 1 to 9 with out 0), for the second number you have 9 options as well (0 to 9 minus the first digit that was already used) and for the third digit you have 8 options left.
So the number of possibilities is 9 x 9 x 8 = 648.


5. The best answer is D.
There are (8 x 9) 72 possibilities of shirts + pants. (2 x 3) 6 Of the combinations are not allowed. Therefore, only (72 – 6) 66 combinations are possible.


6. The best answer is A.
First digit is 1, the second is 2, the third can be (7,8,9), the forth can be (3,6,9), the fifth and the sixth are dependent with one another. The fifth one is 3 times bigger than the sixth one, therefore there are only 3 options there: (1,3), (2,6), (3,9).
All together there are: 1 x 1 x 3 x 3 x 3 = 27 options.


7. The best answer is C.
The probability of picking the first jar is ½, the probability of picking up a white ball out of jar A
Is 3/(3+2) = 3/5. The probability of both events is 1/2 x 3/5 = 3/10.


8. The best answer is B.
The probability for the first one to be black is: 4/(4+6) = 2/5.
The probability for the second one to be black is: 3/(3+6) = 1/3.
The probability for the third one to be black is: 2/(2+6) = 1/4.
The probability for all three events is (2/5) x (1/3) x (1/4) = 1/30.




9. The best answer is B.
Let Z be the probability of breaking the jar, therefore the probability of both events happening is Z x (1/X) = (1/Y). Z = X/Y.


10. The best answer is B.
The probability of the coin is independent on its previous outcomes and therefore the probability for “head” or “tail” is always ½.


11. The best answer is A.
The total number of cards in the new deck is 12 +52 = 64.
There are (4 + 12 = 16) cards with the number 10.
The probability of drawing a 10 numbered card is 16/64 = 1/4.


12. The best answer is E.
After taking out 3 balls there are 15 left. 15/5 = 3 blue balls is the number of left after we took out 3 therefore there were 6 in the beginning.


13. The best answer is D.
The number of green and yellow balls in the box is A+2.
The total number of balls is 4A +8.
The probability of taking out a green or a yellow ball is (A+2)/(4A+8)=1/4.


14. The best answer is D.
Indicate A as the probability of Michael passing the driving test.
The probability of Sam passing the test is 1/4, the probability of both events happening together is 1/6 so: 1/4 x A = 1/6 therefore A = 2/3.


15. The best answer is B.
Indicate A as the probability of drawing a white ball from the jar.
The probability of drawing a red ball is 1/5.
The probability of drawing both events is 1/10 so, 1/5 x A = 1/10.
Therefore A = ½.


16. The best answer is C.
The basic principle of this question is that one person can’t be elected to more than one part, therefore when picking a person for a job the “inventory” of remaining people is growing smaller.
The probability of picking a girl for the first job is 4/10 = 2/5.
The probability of picking a girl for the second job is (4-1)/(10-1) = 3/9.
The probability of picking a girl for the third job is (3-1)/(9-1) = 1/4.
The probability of all three events happening is: 2/5 x 3/9 x ¼ = 1/30.


17. The best answer is B.
When rolling two dice, there are 36 possible pairs of results (6 x 6).
A sum greater than 10 can only be achieved with the following combinations: (6,6), (5,6), (6,5).
Therefore the probability is 3/36 = 1/12.


18. The best answer is E.
The gender of the first-born is insignificant since we want all children to be of the same gender no matter if they are all boys or girls.
The probability for the second child to be of the same gender as the first is: ½. The same probability goes for the third child. Therefore the answer is ½ x ½ = ¼.


19. The best answer is D.
The coin is tossed three times therefore there are 8 possible outcomes
(2 x 2 x 2). We are interested only in the three following outcomes:
(0,1,1), (1,0,1), (1,1,0).
The probability requested is 3/8.


20. The best answer is A.
We want to make a 3-flower garlands, each should have three colors of flowers in it.
There are two different types of blue and two different types of red.
The options are (2 blue) x (2 red) x (1 yellow) = 4 options.

[smvo]

Hi,

My guesses are these:

5)

All possibilities - restrictions

(8C1*9C1) - (2C1*3C1) = 66

6)

1;2; 7-8-9;3-6-9; (3-1 OR 6-2 OR 9-3)

1*1*3C1*3C1* (1+1+1) = 27

7) b

8)

4/10 x 3/9 x 2/8 = 1/30

9)

1/X*P(b) = 1/y
p(b) = X/Y

[smvo]

I failed the 7 Im not feeling confident for the exame!

[skawal]

3. Three people are to be seated on a bench. How many different sitting arrangements are possible if Erik must sit next to Joe?

a) 2.
b) 4.
c) 6.
d) 8.
e) 10.

i am confused about the problem #3.if Erik must sit next to joe then i think there would be two arrangements.these are

(X,J,E),(J,E,X) where X is another person.Oldman, you consider that erik and joe exchange their position but the question gives you the condition that erik must sit next to joe, not before.

am i right? if not then explain it to me.
please also explain #18.

[Gabe_logan]

these are cool gre level questions indeed! i'll lay my hands on them when i'v got time. Thanks Oldman

[newclearbomb]

Probability is my weak spot, i hope to try these out...

[Gabe_logan]

skawal the trick is to notice that eric and joe can sit on only two spots relative to the third person X( i.e at either side of X), and can also sit on two places relative to themselves.
hence we have XEJ, XJE, EJX, and JEX