Friday puzzle -- A dicey one, part 2

If I was a bettin' man

Sory I missed out on last week's puzzle.

Just looking at the combinatorics, your chance of a winning combination are 'unlucky' in a couple of ways. But you need to consider all the terms of the bet to calculate the expected payout.

Re: If I was a bettin' man

Go ahead and post it. I think we are all alone on this one, too.

Gut feel is yes.

I assume that I understand the problem correctly, and even though he has a higher probability of being a winner, it's not enough to cover th 50% premium on the bet.

S

Re: If I was a bettin' man

As before, there are 216 possible combinations of 3 dice. 13 of those are winning combinations and 14 are losers.

Despite those losing odds, the $0.50 premium on the bet swings the expected return back in your favor:

((13 * $1.50) - (14 * $1.00)) / 216 = $0.025

No :-(

Look again at the bet proposed in part 2 versus the question in part 1.

Re: No :-(

Is it related to the fact that in the first question, it was whether the other two matched the first, and in the second it is that any two match the third?

Re: No :-(

Exactly. And surprisingly, that changes the odds.

I meant "yes :)"

Solution -- A dicey one, part 2

Dan spotted the subtle but very meaningful difference between parts 1 and 2. In part one, we asked for the probabilities that the second and third dice thrown would add or multiply to the first. In part two, the guy offering the bet was suggesting a game to see if the three dice could be made to form a valid addition or multiplication. Surprisingly, that changes the odds significantly. In the first case, there are 15 ways of forming a valid addition versus 14 for product, and removing the 4-2-2 tie, we have 14:13 odds, nearly a tie.

In the second game, there are 45 ways to form a valid addition, and only 25 ways to form a valid product. Removing the three ties gives odds of 42:22, nearly a 2:1 bias toward the addition.

To better see what is happening, consider again three different colored dice (G,B, and R), and the cases where the "target" is three.

In the part 1 version, where the green die is cast first, there are two solutions for product:

G3B3R1 and G3B1R3

and two for sum:

G3B2R1 and G3B1R2

In the part 2 version, there is one additional solution for product:

G1B3R3

and four additional solutions for sum:

G2B1R3 G2B3R1 G1B2R3 and G1B3R2

The same table from part 1 could be updated with new counts, with that count being 1 if it is a triple roll, 3 for a double, and 6 if there are three distinct numbers. The result is the 45 and 25 counts given above.

Clarification

Is it related to the fact that in the first question, it was whether the other two matched the first, and in the second it is that any two match the third?

Dan, you hit the nail on the head there, but I do want to clarify one point lest anyone be confused. By "third" above, you meant what is left after choosing "any two", not the third die thrown. Your "choosing any two" makes your intent clear to me, but I wanted to avoid confusion, since if anyone read it as the first two adding to the third thrown die, we are back to the Part 1 version.

Good thing I'm not a gambler

Here I was expecting to make a killing at an expected return of two and a half cents per throw. Instead I would be losing $0.04167.

Thanks for the question.

Re: Solution -- A dicey one, part 2

I knew what was different, but was not smart enough to actually solve it ;-)

This is similar to the problem that says for a random group of people, hoe many do you need to have over a 50 % probability that two of them have the same birthday? (month and day)

Missed the subtlety

However, I do have a good rule of thumb- If someone comes up to me with a bet, chance is always on their side!

Unless I have some way to alter the outcomes and influence or change them, I will never take the bet. I am trying to teach my kids this as well, and my son still falls for it almost every time...

Steve

And an excellent rule of thumb it is

Put another way, "Never bet against the house".

Vegas was built on that!