Posts tagged Math

The Math of The Shire

Jun04
2008 1 Comment

My brother and his wife and 2 kids — the residents of The Shire — have a rare birthday condition. Rowland is a Leo, Shannon is a Taurus. One of the boys, Roman, is a Leo; the other, a Taurus.

What are the odds of that?

Of what? It’s crucial to write out the problem first. Let’s talk this through and establish facts.

The are 12 zodiac signs. Every person has one and only one sign.

There are 2 pairs of people. The first pair consists of the sign Leo and the sign Taurus. The second pair consists of a Leo and a Taurus.

What are the odds that both Rowland and Roman are Leo?
1/12 * 1/12
What are the odds that both Shannon and John are Taurus?
1/12 * 1/12

(Notice how we didn’t look for the odds of Roman being a Leo BUT NOT a Taurus…
Or of John being a Taurus BUT NOT a Leo…
That would be redundant. The 1/12 handles it already. Being a Leo excludes Roman from being a Taurus. )

So the answer to the above problem is: 1/12 * 1/12 * 1/12 * 1/12
Those are VERY high odds. 4.74 e-05

But what if we look instead at the Kind of Arrangement rather than the specific signs? I think this is the question being asked:

The are 12 zodiac signs. Every person has one and only one sign.
There are 2 pairs of people. The first pair (parents) consists of 2 different signs. The second pair’s signs match up with the first pair.

Let’s take that one at a time:
1. What are the odds that 2 people would not have the same zodiac sign?
A: Set the sign of person 1 to a value, X, as a constant. (X in this case can be “Leo”). The constant is equal to 1, because there was a 100% chance person 1 would have a sign.
So, what are the odds that person 2 is not X?
A: 11/12 (Shannon had an 11 out of 12 chance of not being a Leo)
She could have been any number of non-Leo signs: Aries, Pisces, Sagittarius….. it doesn’t matter. But for the purpose of Step 2, we have to define a sign. Let’s go with Taurus.

2. In the second pair (children), what are the odds that person 1 is a Leo and person 2 is a Taurus?
A: We’ve already did this one…. it’s 1/12 * 1/12

The math is: 1 * 11/12 * 1/12 * 1/12
.917 * .083 * .083

The final answer is: .0063
Or, in English:
There was a 63/10,000 chance….

By the way, when I was trying to work this out, a Camp Happy citizen suggested I’d be good at poker. However, he was assuming I use Numbers and Probability to make decisions. Alas, as a Myers-Briggs INFP, I too often go by “instinct” and “feelings.” I’m trying to change that, though! What are the odds of me succeeding?

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The Math of Camp Happy

May02
2008 Leave a Comment

Camp Happy has its own problems which require Camp Happy-style solutions. Take toasting, for instance. There’s a lot of drinking, and celebrating, and with that comes toasts. But just how many clinks will result from a toast, when N is the number of participants, and each participant must clink each other’s glass?

My sister Christine posed this problem a year or two ago; long before I arrived. There was 5 at a table, including cousin A.J., who quickly offered:
4 + 3 + 2 + 1 = 10. Correct outcome.

Or put another way: x = (N-1) + (N -2) + (N-3) + (N-4) ….. keep going until (n-(n-1)).

Christine later devised a simpler formula:

x = N * (N-1) / 2

Example: Three people want to toast the fact that someone just bought a $120 annual state parks pass, and therefore gains admission to all the awesome state beaches for his or her carload.
2 clinks will be necessary to provide for each glass clinking every other glass. Does that # seem low? Well, remember: each clink brings 2 of the 3 glasses together.

I have looked into the matter and deducted that:
For any number of glasses (x) with clinks (y), the addition of 1 glass will add x to clinks.
That is: new clinks = x + y.
If you have 8 glasses, then 28 clinks are needed. If one more person joins in, then 36 clinks are needed (28 + 8).

Try it for yourself. Pick up some fine wine at your local wine shop. There are so many tasty wines available at a low cost. The glut of grapes hasn’t yet been overcome by commodity-inflation. Cheers to that!

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