50 Coins Add to a Dollar
Let's say I have 50 coins in my hand that add up to exactly $1. One coin falls out. What is the chance that it is a penny?
Answer
We only allow pennies (1¢), nickels (5¢), dimes (10¢) and quarters (25¢). There are exactly two ways to make 50 coins total $1. You only need to remember two numbers, 40 and 45:
\[
\begin{aligned}
&\text{(1) }40\text{ pennies},\;8\text{ nickels},\;2\text{ dimes};\\
&\text{(2) }45\text{ pennies},\;2\text{ nickels},\;2\text{ dimes},\;1\text{ quarter}.
\end{aligned}
\]
In case (1) the fraction of pennies is \(40/50=0.80\); in case (2) it is \(45/50=0.90\). Hence without further information the probability must lie between 0.80 and 0.90.