# My Favorite One-Liners: Part 79

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

I’ll use today’s quip when there are multiple reasonable ways of solving a problem. For example,

Two fair dice are rolled. Find the probability that at least one of the rolls is a six.

This can be done by directly listing all of the possibilities:

$11 \qquad 12 \qquad 13 \qquad 14 \qquad 15 \qquad 16$

$21 \qquad 22 \qquad 23 \qquad 24 \qquad 25 \qquad 26$

$31 \qquad 32 \qquad 33 \qquad 34 \qquad 35 \qquad 36$

$41 \qquad 42 \qquad 43 \qquad 44 \qquad 45 \qquad 46$

$51 \qquad 52 \qquad 53 \qquad 54 \qquad 55 \qquad 56$

$61 \qquad 62 \qquad 63 \qquad 64 \qquad 65 \qquad 66$

Of these 36 possibilities, 11 have at least one six, so the answer is $11/36$.

Alternatively, we could use the addition rule:

$P(\hbox{first a six or second a six}) = P(\hbox{first a six}) + P(\hbox{second a six}) - P(\hbox{first a six and second a six})$

$= P(\hbox{first a six}) + P(\hbox{second a six}) - P(\hbox{first a six}) P(\hbox{second a six})$

$= \displaystyle \frac{1}{6} + \frac{1}{6} - \frac{1}{6} \times \frac{1}{6}$

$= \displaystyle \frac{11}{36}$.

Another possibility is using the complement:

$P(\hbox{at least one six}) = 1 - P(\hbox{no sixes})$

$= 1 - P(\hbox{first is not a six})P(\hbox{second is not a six})$

$= 1 - \displaystyle \frac{5}{6} \times \frac{5}{6}$

$= \displaystyle \frac{11}{36}$

To emphasize that there are multiple ways of solving the problem, I’ll use this one-liner:

There are plenty of ways to skin a cat… for those of you who like skinning cats.

When I was a boy, I remember seeing some juvenile book of jokes titled “1001 Ways To Skin a Cat.” A recent search for this book on Amazon came up empty, but I did find this:

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