Interesting Probability

Probability Level 1

A real number $x$ is chosen randomly and uniformly from the interval $[2,10].$ What is the probability that the greatest integer less than or equal to $\frac{x}{2}$ is even?

The answer is 0.5.

2 solutions

Aditya Joshi
Feb 16, 2014

First, let's note that $\lfloor \frac{x}{2}\rfloor$ will be $1,2,3,$ or $4$ with probability 1.

Let's back out what $x$ give $\lfloor\frac{x}{2}\rfloor = 2.$ For this, we need $4\le x < 6,$ an interval of length 2.

Let's back out what $x$ give $\lfloor\frac{x}{2}\rfloor = 4.$ For this, we need $8\le x < 10,$ an interval of length 2.

Thus, via 1-dimensional Geometric Probability , the probability that it is even is $\frac{2+2}{10-2} = \frac{4}{8} = 0.5.$

I have done this differently and the results are different.

For an interval [2,10] we have 9 real numbers to choose from for x.

Now we need to respect the rule that the highest integer that is less or equal to x/2 is also even. This is true for when x is [9;8;5;4].

As such the probability of the rule is 4/9.

What did I do wrong here?

Miguel Melo - 5 years, 1 month ago

I did EXACTLY the same lol and I don't really get the solution

Vincent Gauvin - 4 years, 7 months ago

The problem states REAL NUMBERS not INTEGERS. There are only 9 integers to choose from, but infinite real numbers, such as 3.678768 or Euler's number or Pi...So, You have to use Geometric Probability.

Aaghaz Mahajan - 4 years, 2 months ago

What you (and I) forgot was that 9.999 is a possible value for "x" that would result in an even number by the stated formula. Once you realize that the problem is referencing 2 different types of numbers (whole and real) you also see that the probability of pucking x=10 is close enough to zero that the resulting total probability is .5

John Turner - 4 years, 7 months ago

It is actually [2,10], which means 5 is also a possibility, so the chances would be 4/10, or 0.4

Vatsal Trivedii - 4 years, 1 month ago

The probability of $x=10$ is exactly zero so the above answer is correct: 1,2,3,4 have probability 1. Probability zero does not mean it's impossible.

Miguel B - 2 years, 5 months ago

Maybe this can be useful. First let's convert this model from choosing x to choosing x/2. Now x/2 can take values in [1,5]. But we need those values where the greatest integer ≤ x/2 is even and there are only two even integers in [1,5] i.e. (2 and 4). Draw a line of x/2 with points [1,5] and we can see that if we choose points between 2-3 and 4-5 then only the greatest integer can be 2 or 4. Hence, P = 2 units/ 4 units = 0.5

Amit Kumar - 9 months ago

Advaith Kumar
May 15, 2020

Its simple! notice that for every odd number between 1 and 10 inclusive, $floor(x/2)$ is odd => P = 5/10 = 1/2

Interesting Probability

The answer is 0.5.

2 solutions

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