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Set A consists of 25 distinct numbers. We pick n numbers [#permalink]
10 Jan 2010, 23:29

Expert's post

00:00

A

B

C

D

E

Difficulty:

35% (medium)

Question Stats:

54% (02:02) correct
45% (01:00) wrong based on 48 sessions

This is my question, so no OA just my solution.

Set A consists of 25 distinct numbers. We pick n numbers from the set A one-by-one (n<=25). What is the probability that we pick numbers in ascending order?

(1) Set A consists of even consecutive integers; (2) n=5.

Re: Probability of picking numbers in ascending order. [#permalink]
11 Jan 2010, 01:15

Expert's post

shalva wrote:

Bunuel wrote:

So, if B is the correct answer what's the probability then?

I've absolutely no idea

Somehow the probability depends on the first number we pick and changes with each next pick

let's suppose our Set consists of numbers x1, x2 . . . x25, where numbers are sorted in ascending order.

Probability of picking x1 and x21 as first number is the same. But - probability of picking 5 number in ascending order beginning with x1 is much much higher then beginning with x21

That is not so. But the issue you mentioned is the key part to answer the question.
_________________

Re: Probability of picking numbers in ascending order. [#permalink]
11 Jan 2010, 15:45

1

This post received KUDOS

I think B is the right answer

this is my reasoning for the actual probability

Given 5 nos (doesn't matter what they are) for the first choice- there is exactly one option out of 5 options. i.e we should pick the smallest of the nos. so probability is 1/5 for the second choice - again there is exactly one option i.e smallest no of the remanining for nos. so probability is 1/4 for the 3rd choice- only one smallest no of the remaining 3 nos. so probability is 1/3

Re: Probability of picking numbers in ascending order. [#permalink]
11 Jan 2010, 16:06

Expert's post

janani wrote:

I think B is the right answer

this is my reasoning for the actual probability

Given 5 nos (doesn't matter what they are) for the first choice- there is exactly one option out of 5 options. i.e we should pick the smallest of the nos. so probability is 1/5 for the second choice - again there is exactly one option i.e smallest no of the remanining for nos. so probability is 1/4 for the 3rd choice- only one smallest no of the remaining 3 nos. so probability is 1/3

and so

so total probability is 1/5*1/4*1/3*1/2*1 = 1/120

+1.

We should understand following two things: 1. The probability of picking any n numbers from the set of 25 distinct numbers is the same. For example if we have set of numbers from 1 to 25 inclusive, then the probability we pick n=5 numbers {3,5,1,23,25} is the same as that of we pick n=5 numbers {9,10,4,6,18}. So picking any 5 numbers \{x_1,x_2,x_3,x_4,x_5\} from the set is the same.

2. Now, imagine we have chosen the set \{x_1,x_2,x_3,x_4,x_5\}, where x_1<x_2<x_3<x_4<x_5. We can pick this set of numbers in 5!=120 # of ways and only one of which, namely \{x_1,x_2,x_3,x_4,x_5\} is in ascending order. So 1 out of 120. P=\frac{1}{n!}=\frac{1}{5!}=\frac{1}{120}.

According to the above the only thing we need to know is the size of the set (n) we are choosing from the initial set A.

Re: Probability of picking numbers in ascending order. [#permalink]
30 Aug 2010, 04:05

If we pick n numbers, probability of picking in ascending order will always be [1/{(25*24*...(25-n)}]. Explanation: No of ways of picking n numbers from 25 (25*24*...(25-n) .. And out of that in only one all will be in ascending order. So probability will be [1/{(25*24*...(25-n)}]

Re: Probability of picking numbers in ascending order. [#permalink]
22 Sep 2010, 05:27

let the numbers be a1 , a2 , a3 , ............, a25

Clearly option one is insufficient...Now as per option 2 (n=5) There are 25C5 ways to select a set of 5 different numbers. Now if we consider all the permutations of these 5 diff numbers , then only one satisfies our criteria . Therefore out of 5! cases , only 1 is favorable and hence the probability is 1/5! = 1/120.

Therefore , the ans is B according to me.

had it been some other n<=25 , the probability would be 1/n!

Re: Probability of picking numbers in ascending order. [#permalink]
28 Sep 2010, 14:45

1

This post received KUDOS

How about this We have a set consists of 6 numbers {1,2,3,4,5,6} and the probability that we pick 3 numbers in ascending order : If I use your approach I will get [1][/3] * [1][/2] = [1][/6] but check it out {1,2,3},{1,2,4},{1,3,4},{1,4,5},{2,3,4},{2,5,6},{3,4,5}...........

Re: Probability of picking numbers in ascending order. [#permalink]
28 Sep 2010, 23:04

Expert's post

cagdasgurpinar wrote:

How about this We have a set consists of 6 numbers {1,2,3,4,5,6} and the probability that we pick 3 numbers in ascending order : If I use your approach I will get [1][/3] * [1][/2] = [1][/6] but check it out {1,2,3},{1,2,4},{1,3,4},{1,4,5},{2,3,4},{2,5,6},{3,4,5}...........

1/6 would be a correct answer for your example: if you continue to write 3 numbers sequences in ascending order from a set {1, 2, 3, 4, 5, 6} you'll get 20 possibilities and total # of picking 3 numbers from 6 when order matters is P^3_6=120 --> P=\frac{20}{120}=\frac{1}{6}.

Let's consider smaller set {1, 2, 3, 4}. What is the probability that we pick 3 numbers in ascending order?

P=Favorable scenarios/Total # of possible scenarios.

# of favorable scenarios is 4: {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, ; Total # of possible scenarios is 24: P^3_4=24;

P=\frac{# \ of \ favorable \ scenarios}{Total \ # \ of \ possible \ scenarios}=\frac{4}{24}=\frac{1}{6} or P=\frac{1}{3!}=\frac{1}{6}.
_________________

Re: Probability of picking numbers in ascending order. [#permalink]
02 Oct 2010, 11:42

According to my understanding, probability for option b can be calculated by: As we have to choose 5 among 25 so 25c5....(1) then we can arrange those five in 5! ways.. so the outcome will be 25c2 * 5! = 25p5. so finally out of all sets only 1 will be in ascending order, so ans = 1/(25p5) plz correct me if i m somewhere wrong..

Re: Probability of picking numbers in ascending order. [#permalink]
02 Oct 2010, 12:07

Expert's post

sudhanshushankerjha wrote:

According to my understanding, probability for option b can be calculated by: As we have to choose 5 among 25 so 25c5....(1) then we can arrange those five in 5! ways.. so the outcome will be 25c2 * 5! = 25p5. so finally out of all sets only 1 will be in ascending order, so ans = 1/(25p5) plz correct me if i m somewhere wrong..

No, P=\frac{1}{n!}=\frac{1}{5!}=\frac{1}{120} (please see the solution above).

If we do the way you are proposed then:

Total # of outcomes = P^5_{25} - total # of ways to pick any 5 numbers out of 25 when order matters; Favorable outcomes = C^5_{25}.