### Probability Question

**Question: **

[x] = greatest integer less than equal to x. If x lies between 3 and 5. What is the probability than [x^{2}] = [x]^{2}?

A. Roughly 0.64

B. Roughly 0.5

C. Roughly 0.14

D. Roughly 0.36

**Correct Answer: ****(C)**

**Explanation:**** **

Let us take a few examples.

[3^{2}] = [3]^{2}

[3.5^{2}] =12 [3.5]^{2} = 9

[4^{2}] = 16 [4]^{2} = 16

For x Î (3, 5). [x]^{2} can only take value 9, 16 and 25.

Let us see when [x^{2}] will be 9, 16 or 25

If [x^{2}] = 9, x^{2} [9, 10)
=> x[3,

)

[x^{2}] = 16 => x^{2} [16, 17)
=> x [4, )
In the given range [x^{2}] = 25 only when x = 5

So [x^{2}] = [x]^{2} when x [3, ] or [4, ) or 5.

Probability =
= = 0.14

**Difficulty Level 2**

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