If a randomly selected non-negative single digit integer is

Question

If a randomly selected non-negative single digit integer is added to {2, 3, 7, 8}. What is the probability that the median of the set will increase but the range still remains the same?

A. 0.2
B. 0.3
C. 0.4
D. 0.5
E. 0.6

Bunuel · Accepted Answer

If a randomly selected non-negative single digit integer is added to {2, 3, 7, 8}. What is the probability that the median of the set will increase but the range still remains the same?

A. 0.2
B. 0.3
C. 0.4
D. 0.5
E. 0.6

" A randomly selected non-negative single digit integer  is added to {2, 3, 7, 8}":
We are selecting from non-negative single digit integers, so from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. These 10 digits represent the total number of outcomes.

Hence,  the total number of outcomes is 10 .

We need to find the probability that the median of the set will increase but the range still remains the same.

The median of {2, 3, 7, 8} is (3 + 7)/2 = 5 --> the number selected must be greater than 5
The range of {2, 3, 7, 8} is 8 - 2 = 6 --> the number selected must be from 2 to 8, inclusive.

To satisfy both condition the number selected must be 6, 7, or 8.

Hence,  the number of favorable outcomes is 3 .

P = (favorable)/(total) = 3/10.

Answer: B.

Hope it's clear.

fluke · Answer

Non-negative digit adds to the confusion.

A digit is always non-negative.

Condition will be satisfied if the digit is one of: {6, 7, 8}

Total digits=10

P=3/10=0.3

Ans: "B"

Marcab · Answer

what about 4 and 5?
suppose if i add 4, then also the median of the set increases whereas the range remains the same.

fluke · Answer

Median is the average of two middle terms if there are even number of elements arranged in ascending order.
Median is the the middle terms if there are odd number of elements arranged in ascending order.

Here, there are even number of elements.
{2, 3, 7, 8}
Middle terms are 3 and 7.
Their average: (3+7)/2=10/2=5
So, median is presently 5.

If you add 4 and 5, medians will be 4 and 5 respectively.
{2, 3, 4, 7, 8}: Middle term=4(odd number of elements)->Ignore
{2, 3, 5, 7, 8}: Middle term=5(odd number of elements)->Ignore
{2, 3, 6, 7, 8}: Middle term=6(odd number of elements)
{2, 3, 7, 7, 8}: Middle term=7(odd number of elements)
{2, 3, 8, 7, 8}={2, 3, 7, 8, 8}: Middle term=7(odd number of elements)

tranglenh · Answer

Call X - new number
Let's consider the 2 conditions:

* median increase:
Old median : 2+3+7+8/4 =5
New median: 2+3+7+8+x/5 = 20+x/5
New median increase: 20 + x/5 > 5 --> x >5
* Range unchanged:
with the same smallest number is 2, x>5 then new x must <= 8
---> 5<x<=8 then x can be (6,7,8)
* Probability = 3/10 = 0.3

LalaB · Answer

sorry for a stupid question )) but how did u get 10? :oops:

sorry

fluke · Answer

non-negative digits: {0,1,2,3,4,5,6,7,8,9}. How many of these are there? 10, right.

Probability=Favorable Outcomes/Possible outcomes and "Possible outcomes"=10 because any of the 10 digits can possibly be chosen.

DeeptiM · Answer

Isn't "0" neither negative nor positive??

Shouldn't it then be 3/9??

fluke · Answer

0 is indeed neither negative nor positive. But, non-negative means everything that's not negative AND 0 is not negative. So, we must consider 0 for all the non-negative cases. We must consider 0 for non-positive case as well because 0 is not positive.

Bunuel · Answer

We are selecting from non-negative single digit integers, so from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. These 10 digits represent the total number of outcomes.

The part about the range and the median are constraints, which limit/define the favorable outcomes.

Hope it's clear.

reddevils · Answer

Bunuel - Ok but still I unable to get my head around the range constraint. The question talks about the range of single digit non negative numbers? I thought the question talks about the range of the set of 4 numbers given ({2, 3, 7, 8}).

Please clarify.

Please clarify.

reddevils · Answer

Thank you Bunuel. I really do feel numb. Don't know what I was thinking regarding the constraint of range. Thanks again for taking the pains to clarify a dumb query.

Cheers to you!!

Bunuel · Answer

There is nothing confusing there.

The median of {2, 3, 7, 8} is (3 + 7)/2 = 5.
The range of {2, 3, 7, 8} is 8 - 2 = 6.

If you add 3, then the new set is {2, 3, 3, 7, 8}.

The median of {2, 3,  3 , 7, 8} is 3. So, the median in this case does NOT increas.

Archit3110 · Answer

total non -ve intgers single digit are ( 0,1,2,3...9) ; 10
and median of the set given 2,3,7,8 ; 3+7/2 ; 5
range of set is 8-2 ; 6
we need an integer value such that the median of set increases but range is same 
so possible integer which can be included to get this condition not satisfied ; ( 0,1,2,3,4,5,9) ; 7 
1-7/10 ; 3/10 ; .3
IMO B ; 
or say we have 3 possible digits ; 6,7,8 ; 3/10 ; .3
IMO B

piyush348 · Answer

Hi,
 Please explain why did we not take total no. of cases as 8 . As there are 8 ways where range is not changing. I am struggling with probability, this might help me understand for the next time.

Bunuel · Answer

The total number of possible outcomes is 10, because we're choosing from all non-negative single-digit integers, which are 0 through 9.

You're referring to the 8 values (2 to 9) where the range might stay the same — but that’s related to favorable outcomes, not the total number of possibilities.

In probability, the denominator (total cases) includes every possible option you're selecting from, regardless of whether they satisfy the condition or not.

pusansen99 · Answer

Wait, I thought we were adding (addition '+') a digit to the existing numbers in the set. Does the question mean adding another item to this set?

Bunuel · Answer

It means adding a new number to the set, not adding to each element. Have you checked the discussion above?

If a randomly selected non-negative single digit integer is

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If a randomly selected non-negative single digit integer is

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