This
weekend my friends and I played a board game called Stone Age, where you hunt,
collect resources and construct buildings.
During the game, a player drew a card that called for rolling dice to
determine what resources we'd get. He
wanted to roll a '6' because that meant he'd get a valuable farm, and so we
talked about the odds of rolling a '6' with four dice.

This
leads us to probability. Probability is
simply the measure of how likely something is to happen. It can be a small thing, like the flip of a
coin, or something big, like the likelihood you'll get into a car wreck.

A
probability can be expressed as a fraction between 0 and 1, or as a percentage
of 0% to 100%. In our daily lives, we
may not take the time to write down the exact fractions or percentages, but we
make educated guesses all the time.

If
you're choosing what route to take home from work and you know it's the Friday
before a three-day weekend, you'll predict what highways or bridges will be jammed,
and then choose a different road. If
you're in an office pool for the NCAA basketball tournament, you'll figure some
rough probabilities in your head as you pick your bracket winners.

We
do this stuff all the time, so we shouldn't let the math part scare us.

Since
the 1500s, mathematicians and gamblers have studied probability. Famous thinkers like Fermat and Pascal
examined this topic, and there has long been a connection between betting and
math. Even the phrase "games of
chance" tells us they involve probability.
One of the early papers on probability, titled "Doctrine of
Chances" by DeMoivre, contained problems which calculated the odds of
winning lottery tickets.

One
odd branch of this area is actuarial science, where insurance actuaries use
math and statistics to make probability predictions about all sorts of things,
including death. An early example of
this is the London Life Table created by John Graunt in the 1600s. Graunt used the lifespans of 100 people in
the city and noted how long they lived, which provided useful data about the
appalling mortality rate among children of that era.

If
you're curious how many more years you may live, follow this link to a Social
Security chart. The chart does not take
into account where you live, or if you're in a dangerous occupation, etc., so
it is quite broad. But you can look at
your gender, figure out what age you were in 2007 (when the chart was made),
and then look at the column labeled "Life Expectancy" and see how
many more years you may live. It is
sobering.

And
a reminder that you should use well the days you have on this Earth.

(Sources include: Education World, an article by Amy Troutman
at Wichita.edu, and Social Security Online.
The pic is from Google Images.)

So back to the question, what is the probability of rolling a six with four dice? (Answer: 51.8%, more in a minute) It’s kind of strange but the process actually entails finding the probability of NOT rolling a six.

ReplyDeleteFor one die it is easy 1/6. Or the odds of NOT rolling a six are 5/6. Then you can write 1 – 5/6 = 1/6.

The probability of NOT rolling a six with two dice is actually 5/6 X 5/6 = 25/36. Then write 1 – 25/36 = 11/36, which are the odds of rolling a six with two dice. NOTE: It is incorrect to assume that the odds of rolling a six are with two dice is 1/6 + 1/6 = 2/6 even though the percentages are close at this time but won’t be as we add more dice. (Think about it, if you had six dice would the probability be 6/6? Are you guaranteed to roll a six with six dice rolls?)

Three dice: 1 – 125/218= 42.7%

The odds of NOT rolling a six with four dice is 5/6 X 5/6 X 5/6 X 5/6 = 625/1296. So the probability of rolling at least one six with four dice is 1 - 625/1296 = 671/1296 = 51.8%

At first glance you may think you would only need three dice to roll at least one six half the time but in reality you need four dice. The flaw in this thought process is when you roll six dice or more dice it would seem you are guaranteed to roll at least one six. The reality is it is NEVER guaranteed that you will roll a six. In fact if you roll 13 dice you will NOT roll a six 10% of the time.

Probability is intuitive… and it isn’t.

Olaf, thanks for the dice explanation. I should have gone back to that and explained what happened, but I stumbled off onto the Trail of Tangents with the mortality tables.

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