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GMAT Club Olympics 2025 (Day 1): Ann and Bob each randomly select an

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Statement 1 is sufficient since P(A>B)=9/20

Statement 2 is insufficient.

Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

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Answer is option C. Both statements are needed

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To know the probability you only need to know the range of the integers, From there half of the integers will always be more than the other half
S1 Gives us exactly that hence sufficient
S2 No info about X hence insufficient
Ans A

Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

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Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

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1. Y-x =9 -> y=x+9
Lets X=0 -> y=9

Total number of cases in which Ann and Bob select an integer from 0 to 9 is 10*10=100

Total number of cases in which Ann select integer greater than Bob = 9*1(bob selects 0)+8*1+.....0*1(Bob selects 9)
Hence, probability can be estimated.

2. y=-20, nothing given for x=> can't calculate the number of total or favorable cases.

Hence, Answer is A

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(1) y - x = 9

This tells us that we have a range of 9, it could be 1,2...,10 or 11,12..., 20. Lets take one example : 1,2,3..10
Now Bob can select any number from 1 to 10, and we need to see the cases where Ann's number will be greater than Bob's:
If Bob picks 1, Ann can pick any from 2,3..10 > 9 numbers
If Bob picks 2, Ann can pick any from 3,4..10 > 8 numbers
If Bob picks 3, Ann can pick any from 4,5..10 > 7 numbers
...
If Bob picks 9, Ann can pick only 10 > 1 number. If Bob picks 10, any number Ann picks will be smaller or equal to 10, so we don't count that case.

From here, we can get the sum of all cases and divide the total by 100 to get the probability.So knowing the range is enough. As long as the range is same, the sum of all cases will remain the same as well.

Statement 1 is sufficient

(2) y = -20
Here we do not know what x is. As seen from previous statement, we must know the range between y and x to find all the cases where Ann can select a greater number than Bob.
Here x could be -50 or -100, and the probability could vary on x's value.

So Statement 2 is not sufficient

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Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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We have to check if Ann selects an integer which is greater that Bob. To solve this problem, we need to find the set of possible values that can be selected.

Statement 1:
y - x = 9

This gives the range of the integers, since the numbers can only be selected from between them. This would give us a set so we can solve the problem.

For Example,

y=10, x=1
Set of integers=> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Now we have to manually check for each possible value Ann will select:
-> 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
= 45

the actual values if x and y are irrelevent since the probability will be the same.

So Statement 1 is Sufficient

Statement 2:
y = -20

This only gives the upper bound, only using the upper bound cannot help us find the set of values. Which is not enough to solve the problem.

So Statement 1 is Not Sufficient

Answer:
(A)

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Both insufficient, E, theres no way from this info we get to know which integer Bob and Ann select.

Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

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for the GMAT Club Olympics Competition

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IMO, statement I alone is sufficient.
Statement I says y-x = 9, which means the range can be any thing like:
0 =< x,y <= 9
-29 =< x,y <= -20
and anything with difference of 9.
So if the range is 0 to 9, Ann can pick a number greater than bob in (9+8+7+6+......+1) ways out of 10p2 ways, which will give us the probability.
Therefore, answer should be option A.

Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

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for the GMAT Club Olympics Competition

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For St1, Take [1,10] as an example.

Or take (n,n+1,n+2,...........n+10)

P(Ann selected no>bobs selection) = P(Bob chooses n and Ann chooses n+1/n+2/....n+10) + P(Bob chooses n+1 and Ann chooses n+2/...n+10) + so on. = 1/10 x 9/10 + 1/10 x 8/10 +.....1/10 x 1/10.... =

If Bob chooses "1" out of her 10 options, Ann can choose 2 to 10 (9 options out of 10).
If Bob chooses "2", Ann cannot choose 1 or 2, but she can choose 3 to 10 (8 options)
...
If Bob chooses "9", Ann can still choose 10 (1 option)

So Overall P = sum of all these individual possibilities.

Statement 1 is sufficient. Option A.

Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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We do not know what numbers are actually chosen.

Answer: E IMO

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A) tells is the number of integers in the set. eg. 19-10=9 or 11-2=9. Since we know that a digit is randomly picked from 10 digits, we can calculate the Probability.
Sufficient
B) tells us the one end of the set, but does not give any information regarding the number of digits in the set.

Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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We need to identify the probability that integer choosen by A>B = Prob (A>B)

Statement 1: Let y=10; x=1; y-x=9

Let pairs of numbers choosen by (A,B) be (1,1), (1,2).........(9,9), a total of 81 pairs.

Out of this, 36 pairs are such that (A>B), eg (2,1), (3,2)....(6,5)....(9,8). Make sure to count out cases where they have picked the same numbers

Hence, Prob (A>B) = 36/81 = 0.45 (approx)

=> Statement 1 is Sufficient, eliminate BCE

Statement 2: y=-20.

We dont know anything about x, hence we will not be able to determine a range or possible values.

=> Statement 2 is insufficient, eliminate D

Answer is A

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Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

The answer to this question would be (A) - that statement 1 is sufficient to answer the question but statement 2 alone is not sufficient.

Statement 1 provides us with the "number" of values within the range from x to y by providing the difference of 9. This allows for a probability to be calculated even if we do not need to calculate it for this question. This eliminates answer choices C (both together) and E (neither). Statement 1 is sufficient, however we need to test if statement 2 is sufficient.

Statement 2 provides us with a value of y but it does not tell us anything about x (not even how many values away it is). We would need statement 1 in conjunction to figure out the probability with the range of 9 to figure out probability. However alone, statement 2 is not enough and we can eliminate answer choice B. Answer choice C can also be eliminated since Statement 1 is enough to calculate probability and Statement 2 is not needed as well.

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Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

This question was provided by GMAT Club
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Let's consider some cases to make a sequence ( Ann selects is greater than the one Bob selects ):
1. Let's consider Bob selects y, the highest integer, then Ann can select x to y-1 integer ( y-1-x+1 = y-x integers)
2. Let's consider Bob selects y-1, the second highest integer, then Ann can select x to y-2 integer ( y-2-x+1 = y-x-1 integers)

And so on, the total probability would be: = ((y-x) + ( y-x-1) + (y-x-2 ) .......)/(y-x+1)^2

Now, from 1, we can get the value of (y-x), the only unknown variable; hence, Statement 1 is sufficient.

From 2, we can only get the value of y, not the y-x; hence, Statement 2 is insufficient.

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Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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The question does not state that x and y are positive. So (1) gives multiple option for y-x. (2) doesn't give x. Together we get both and X and Y. Which means c.

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We need to know either x and y or the number of integers in between x and y, calculate the probability that one integer is greater than the other.
Statement 1: It can be imagined that x is 9 units to the left of integer y.
That is y=x+9, that means x, x+1, x+2, x+3, x+4, x+5, x+6, x+7, x+8, y are all there.
So the probability can be calculated. So sufficient.
Statement 2: y =-20, no detail about x insufficient.
Hence A.

Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

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for the GMAT Club Olympics Competition

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Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

we want to calculate P (A>B), where A is the number Ann chooses, and B is the number Bob chooses.
Due to the principle of symmetry, when choosing from the same set,

The probability that A>B is the same as the probability that B>A (by symmetry).
The probability that A = B is 1/total outcomes

Since all possibilities must add up to 1, and due to symmetry principle above
2P(A>B) + P(A=B) = 1
P(A>B) = (1 - P(A=B))/2

To calculate P(A = B) = 1/total outcomes, total outcomes = y-x+1

1) y - x = 9. We can calculate P(A = B) and P A > B as a result. Sufficient
2) y = -20. we don't know anything about x. Insufficient.

Choose A

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Bunuel

Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

This question was provided by GMAT Club
for the GMAT Club Olympics Competition

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[size=100]We have two people, Ann and Bob, who each pick an integer randomly from the same range of integers, specifically from integer x to integer y, inclusive. We need to determine the probability that the integer Ann picks is greater than the one Bob picks.
We're given two separate pieces of information (statements), and we need to figure out if each one alone is sufficient to answer the question, or if we need both together.
Breaking Down the Probability
First, let's think about how to calculate the probability that Ann's number is greater than Bob's when both are selecting from the same range.[/size]

[size=100]Total Possible Outcomes: If there are n integers in the range from x to y, then:
[/size]
- [size=100]Ann has n choices.[/size]
- [size=100]Bob has n choices.[/size]
- [size=100]Total possible (Ann, Bob) pairs: n×n=n^2.[/size]
[size=100]Favorable Outcomes (Ann > Bob):
[/size]
- [size=100]For Ann's number to be greater than Bob's, if Ann picks a, Bob must pick a number less than a.[/size]
- [size=100]The smallest number Ann can pick is x, and there are no numbers less than x for Bob to pick, so 0 favorable outcomes here.[/size]
- [size=100]If Ann picks x+1 Bob can pick x: 1 favorable outcome.[/size]
- [size=100]If Ann picks x+2, Bob can pick x or x+1: 2 favorable outcomes.[/size]
- [size=100]...[/size]
- [size=100]If Ann picks y, Bob can pick any number from x to y−1: y−x favorable outcomes.[/size]
[size=100]So, total favorable outcomes = 0+1+2+⋯+(y−x) =$ (y-x)*(y-x+1)/2$[/size]

[size=100]Calculating Probability:
[/size]
- Probability PP = Favorable / Total = $((y-x)*(y-x+1)/2 ) / (y-x+1)^2$

Essentially if you know y-x you can calculate probability

Statement 1 Tells you exactly that, so it is sufficient.
Statement 2 just gives you value of y

Final Determination

Statement (1) alone is sufficient to determine the probability.
Statement (2) alone is not sufficient.
Combining both doesn't add more than what (1) provides.

Therefore, the correct answer is:
A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

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Ann and Bob each randomly select an integer from the integers x to y, inclusive. What is the probability that the integer Ann selects is greater than the one Bob selects?

(1) y - x = 9
(2) y = -20

The probability that Ann selects an integer that is greater than Bob's:

N=y-x+1 (which is the total number of integers)

Total ordered pairs=N^2
Ties=(cases where Ann's selection is equal to bob's) is N
The remaining pairs (N^2-N) are split evenly thus giving

P(Ann>Bob)= (N^2-N)/2N^2=(N-1)/2N

Statement 1: N=9+1=10

(10-1)/2x10= 9/20

Thus statement 1 is sufficient.

Statement 2 is insufficient as it tells us only the upper bound.

Thus the answer is A

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