Probability Puzzles with Obsolete Coins

I love math puzzles, and Henry Ernest Dudeney is the foremost puzzlemeister of all time.

Whenever I need a diversion from the pressing day-to-day busyness of life, I like to crack open one of his books. And get humbled. He stumps me as often as not, but I enjoy the challenge.

Today, I’ve got two of his somewhat-easier puzzles from Amusements in Mathematics, dealing with the probability of obsolete coins tossed or pulled from a bag.

Okay, so the penny isn’t obsolete quite yet, even though the US government quit making them. But the “penny” in these puzzles is the old British pence, which went out of date seemingly ages ago, along with all the other non-decimal coins that made old math puzzles so confusing.

Probability puzzles are great for middle school students because the arithmetic itself is simple. The challenge is in figuring out exactly what calculation to do.

Have fun playing math with your kids!

Things to Know

If you haven’t played with probability before, here are a few things you need to know…

How to Count

In the study of probability, we are not counting existing things, but possible things. We can’t know for sure which result will happen, but we can count the results that might happen.

Sometimes, the situation is simple enough that you can make a list or chart of all the possibilities. When that gets too complicated, the fundamental counting principle comes to your rescue.

But you won’t need the fundamental counting principle for these puzzles, so I’ll explain more about it next week.

Probability

We calculate probability as a ratio, a fraction that compares the results we are interested in to all the possible things that might happen:

Probability = Winners/Possibilities

So you need to count two things. First, count all the possible results of your situation. Then, count the results that “win” whatever challenge you are studying.

Odds

The odds of winning your challenge situation are not the same as the probability of winning. Odds are much more confusing because the word is used so many ways.

Odds for, or odds on, is the ratio of the winning possibilities to the losers.

Odds on = Winners/Losers

Another common usage is odds against, or the ratio of losing possibilities to winners.

Odds against = Losers/Winners

Expected Value

In a game involving money or points, the expected value is the average amount you would win per turn in an infinite number of random plays.

We calculate expected value by finding the value of every possible outcome multiplied by the probability that result will occur, and then adding all those amounts together.

In Dudeney’s puzzle, the expected value is the amount you should be willing to pay to play the game. In reality, the man offering you the game will charge more than that, because the one inviolable rule of gambling is that the house always wins. Over the long run, it’s impossible to earn back the amount you pay to play.

Dudeney’s Two Questions in Probability

Dudeney writes…

There is perhaps no class of puzzle over which people so frequently blunder as that which involves what is called the theory of probabilities. I will give two simple examples of the sort of puzzle I mean. They are really quite easy, and yet many persons are tripped up by them.

A friend recently produced five pennies and said to me:

“In throwing these five pennies at the same time, what are the chances that at least four of the coins will turn up either all heads or all tails?”

His own solution was quite wrong, but the correct answer ought not to be hard to discover.

Another person got a wrong answer to the following little puzzle which I heard him propound:

“A man placed three sovereigns and one shilling in a bag. How much should be paid for the permission to draw one coin from it?”

It is, of course, understood that you are as likely to draw any one of the four coins as another.

My Comments

The sovereigns were each worth 20 shillings, each of which was worth 12 pence.

I question Dudeney’s last statement about the coins. Don’t mints normally try to make coins in different sizes and with smooth or ridged edges, so you can identify what’s in your pocket by feel?

Anyway, I prefer to think in terms of dollars, so you can imagine that the bag contains four bills: three $20’s and a single dollar. Perhaps folded into rings or shirts.

Not Quite Solutions

I don’t want to ruin your fun by giving away the answers. If you found an answer, but you’re not sure whether it’s right, you can check for yourself at the Internet Archive (answer 30).

But I will warn you of the most common errors.

For the penny problem, were you careful to consider all the possibilities? There are only six possible combinations of heads and tails, but most of those combinations can be made more than one way. So there are actually more than thirty ways the coins might land.

For the money-in-a-bag challenge, did you remember to take the shilling (or dollar bill) into account, as well as the more valuable coins? According to Dudeney, most people forget that every draw results in a win of some amount, even if it feels like a loss because of the small value.