Solution: 2014 Winter Final - 13

Author: Michiel Smid

Question

Let $A = \{1,2,3,\dots,100\}$. Let $x$, $y$, and $z$ be elements in $A$ that are chosen independently and uniformly at random. What is the probability that $x = y = z$?

(a)

$\frac{1}{{100 \choose 2}}$

(b)

$\frac{1}{100 \cdot 99}$

(c)

$\frac{1}{100}$

(d)

$\frac{1}{100 \cdot 100}$

Solution

$x$ can be any of the 100 elements in $A$: 100

For $y$ to be equal to $x$, there is only 1 favorable outcome out of 100: 1

Similarly, for $z$ to be equal to $x$, there is only 1 favorable outcome out of 100: 1

The probability that $x = y = z$ is the product of the probabilities of each event:

$P(x = y = z) = \frac{100}{100} \times \frac{1}{100} = \frac{1}{100}$

$P(x = y = z) = \frac{1}{100 \cdot 100}$