There are n cards numbered consecutively, from 1 to n. Two cards are

Question

There are n cards numbered consecutively, from 1 to n. Two cards are drawn. What is the probability that the number on the first card is less than the number on the second card?

(1) The two cards are drawn in succession, with replacement.
(2) n = 10.

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GMATinsight · Answer

Question: Probability that first card number is less than second card Number = ?

Statement 1  : The two cards are drawn in succession, with replacement.
Number of card unknown hence
 NOT SUFFICIENT

Statement 2  : n = 10
We don't know whether replacement is allowed or not hence
 NOT SUFFICIENT

Combining the statemets 
Now we have number of card and also the clarity of process of drawing card hence
 SUFFICIENT

The process is not important but for learning sake:  
 Total cases in which we can get both card same = 10 
 Total outcomes = 10*10 = 100 
 Remaining outcomes = 100-10 = 90 
 Out of 90 remaining outcomes, the probability of first card to be less than second is same as the probability of getting first card greater than second i.e.1/2 
 so favorable outcomes = (1/2)*90 = 45

Required probability = 45/100 = 9/20

Answer: Option C

Related Video (Official GMAT Question):

https://www.youtube.com/watch?v=UgEM7-eTv-Y

TestTaker033 · Answer

There are n cards numbered consecutively, from 1 to n. Two cards are drawn. What is the probability that the number on the first card is less than the number on the second card?

(1) The two cards are drawn in succession, with replacement.
(2) n = 10.

But the quantum of probability would change after the first card is drawn. Example, lets suppose n=10 and first card drawn is 7 (c). On replacement, if card numbered 1-6 are drawn, then only we can calculate the probability of 2nd card being less than 1st card.

So, taking the above example, 
n=10, c=7
P(2<1)= 1/10*6/10= 6/100

when c= 8, then this value changes. 
Let me know where I am wrong.

According to me answer should be E.

Bunuel · Answer

You’re missing the point.

Once we combine the statements, we know everything: the number of cards (10) and the method of drawing (with replacement). That is the entire setup. If we can’t answer the question even after knowing both the total sample space and the process, what else do you think is missing?

How can you say it’s still not sufficient when we literally have all the information required to compute the exact probability?

Simplified breakdown for clarity:

Total outcomes = 10 * 10 = 100 
 Equal card cases (same number both times) = 10 
 Remaining = 90 
 Out of those 90, half will have first < second = 45 
 So, required probability = 45 / 100 = 9/20

Aboyhasnoname · Answer

Hi Bunuel,

Please help me explain this..... Doesn't the probability depends on which card is being drawn?

Suppose I draw 1 in the first draw..the probability of the second one being greated than this is 9/10 
While If i Draw 10 in the first draw... The probability of 2nd one being more than it is 0/10 
or else if i draw 9, then the probability of the next card being greater than is is 1/10 
Please help here

Bunuel · Answer

You're right for those specific cases. If you fix the first card as 1, 9, or 10, the probability does vary. However, we're interested in the overall probability. For that, we consider all 100 possible outcomes and use the shortcut shown: out of the 90 unequal pairs, exactly half will have the first card less than the second, giving 45/100.

There are n cards numbered consecutively, from 1 to n. Two cards are

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GMAT Data Sufficiency (DS) Questions

There are n cards numbered consecutively, from 1 to n. Two cards are

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