JEE Mathematics 2026 Question with Solution

Given: The mean and variance of the observations $2, 4, 10, x, 12, 14, y$ are $8$ and $16$ respectively, with $x > y$.

Find: The probability that when two numbers are chosen without replacement from $\{1, 2, 3, x-4, y, 5\}$, the smaller number is less than $4$.

Using the mean,

So,

Hence,

Using the variance formula,

Therefore,

So,

Now solve the system

Put $y=14-x$ in the second equation:

So $x=6$ or $x=8$. Since $x>y$, we get

Now the set becomes

Two numbers are chosen one after another without replacement.

Total ordered outcomes:

Let $A$ be the event that the smaller of the two chosen numbers is less than $4$.

Use the complement. Let $A'$ be the event that the smaller number is at least $4$. Then both chosen numbers must come from $\{4,5,6\}$.

Number of ordered outcomes in $A'$:

Hence,

Therefore,

Therefore, the required probability is $\frac{4}{5}$. The correct option is A.

Given: The unknown values $x$ and $y$ are determined from the mean and variance, and then probability is to be found from the resulting set.

Find: The probability that the smaller of two chosen numbers is less than $4$.

First determine $x$ and $y$ from

and

The valid pair satisfying $x>y$ is

So the set is

Instead of counting directly when the smaller number is less than $4$, count when this does not happen.

If the smaller chosen number is not less than $4$, then both selected numbers must be among

Since selection is without replacement and in order, the number of such outcomes is

while total ordered outcomes are

Thus,

Hence,

Therefore, the correct option is A.