Dice probability

A die is rolled, find the probability that an even number is obtained?

On the Blue die do you mean 1 1 1 7 7 7? You only have two 7s.
Also on the Green die, do you mean 2 6 2 6 2 6? You only have two 6s

this one looks really simple, you don't have to work out all the possibilities seems to me.
but I think you should do your own homework.

Great question.
I know the answer but will keep it to myself for now.

Here is a pdf link to help answer these type of questions.
http://www.madandmoonly.com/doctormatt/mathematics/dice1.pdf
From: Matthew Conroy's page of mathematics
http://www.madandmoonly.com/doctormatt/mathematics/mathematics.htm

Looks to be similar to question #25
current version July 1, 2012

Quote: anaid23

Blue dice has 1 1 1 1 7 7
yellow dice has 4 4 4 4 4 4
green dice has 2 6 2 6 2 2
orange dice has 3 5 3 5 3 5

Four dice with varying numbers on each face. You and three friends will play a simple game where you will each roll one of the dice and the highest number wins. You get first pick from the dice.

Question: Which die should you choose in order to maximize your chance of winning?

I believe the yellow and orange dice have same odds of winning/being the best choice of dice.

blue has two 7s and four 1s

green has four 2s and two 6s

This seems incredibly simple, so perhaps I'm missing something because it seems obvious.

4 die, all 6 sided are rolled. This means there are 1296 combinations that can be rolled. Obviously Blue dice win 432 times, since two of their sides are higher than every other die roll (2x6x6x6). The other colors all win 288 times each, therefore, the answer is :

BLUE

Odds on who wins :

blue 2-1
yellow 7-2
green 7-2
orange 7-2

If we put a 5% vig on the prop, we would offer : blue +185, yellow +327, green +327, orange +327

Quote: shakhtar

Odds on who wins :
blue 2-1
yellow 7-2
green 7-2
orange 7-2

Isn't 7:2 a higher probability than 2:1?

Quote: tupp

Isn't 7:2 a higher probability than 2:1?

no. 2-1 equals a 33.33% chance of occurring, 7-2 equals a 22.22% chance of occurring

odds are a reflection of the chances of something occurring vs. the chances of something not occurring.

I'm assuming you don't do too much gambling, and by the oft chance that you do, you should probably find another hobby.

Quote: shakhtar

no. 2-1 equals a 33.33% chance of occurring, 7-2 equals a 22.22% chance of occurring

odds are a reflection of the chances of something occurring vs. the chances of something not occurring.

I'm assuming you don't do too much gambling, and by the oft chance that you do, you should probably find another hobby.

No need for the the personal comments.

I do not wish to seem pedantic nor start a semantics argument. However, for the ratios to match "the chances of something occurring vs. the chances of something not occurring," would not the ratios actually have to be inverted, to 1:2 and 2:7, respectively?

Furthermore, even if we invert the ratios, they do not match the percentages given. 1/2≠33.33%, and, likewise, 2/7≠22.22%.

On the other hand the percentages do match another ratio: the chances of something occurring vs. the total possibilities. So, 1/(1+2)=33.33%, and 2/(2+7)=22.22%.

Quote: tupp

No need for the the personal comments.

I do not wish to seem pedantic nor start a semantics argument. However, for the ratios to match "the chances of something occurring vs. the chances of something not occurring," would not the ratios actually have to be inverted, to 1:2 and 2:7, respectively?

Furthermore, even if we invert the ratios, they do not match the percentages given. 1/2≠33.33%, and, likewise, 2/7≠22.22%.

On the other hand the percentages do match another ratio: the chances of something occurring vs. the total possibilities. So, 1/(1+2)=33.33%, and 2/(2+7)=22.22%.

good lord. Odds of blue happening are 2-1, 33.33% chance out of 100%. 33.33% vs 66.66.% chance is 2-1.

7-2 is 22.22% out of 100, hence 22.22% vs. 77.78% . What is 77.78 divided by 22.22 ? 3.5. what is 7 divided by 2? 3.5

you can say 7 to 2, or 2 in 9, whatever you prefer, but odds are traditionally quoted as the ratio quoting your payoff of a bet. a 10-1 shot means you win 10 for every 1 you risk.

This is really hard to figure out when everyone have a different opinion :(

Quote: shakhtar

This seems incredibly simple, so perhaps I'm missing something because it seems obvious.

4 die, all 6 sided are rolled. This means there are 1296 combinations that can be rolled. Obviously Blue dice win 432 times, since two of their sides are higher than every other die roll (2x6x6x6). The other colors all win 288 times each, therefore, the answer is :

BLUE

I agree that this is a simple question.
I mis-read the question.
The pdf I linked to, calculated one die vs. one other die as in 2 players selecting one die each from 4 dice total.
I guess it can be used to do the same calculations by doing each step.
Not needed with the dice values given here.


When rolling 4 6-sided dice there are only 6^4=1296 possible outcomes.

since one die does not copy another die's face this is just straight multiplication that shaktar did.

Blue has 2*6*6*6= 432 ways to win (The 2-7s beat the 6 faces of the other 3 dice)
432/1296 = 1/3 prob winning (3/9)

Yellow has 6*4*4*3= 288 ways to win (it can beat 4,4 and 3 faces of the other dice)
288/1296 = 2/9 prob winning

Green has 2*4*6*6= 288 ways to win (2 of its high faces-the 6, it can beat 4,6 and 6 faces of the other dice)
288/1296 = 2/9 prob winning

Orange has 3*4*6*4= 288 ways to win (3 of its high faces-the 5, it can beat 4,6 and 4 faces of the other dice)
288/1296 = 2/9 prob winning

It would take different some math if the same value was on more than one die.

shaktar then just converted the probabilities into odds and gave a money line for each.
He gets the A+ for content and effort ;)

Quote: anaid23

This is really hard to figure out when everyone have a different opinion :(

It's not opinion, Blue is the correct pick. Simple math. Only 1296 permutations to calculate.

Blue wins 432, Yellow, Orange, and Green win 288 each.

Think of it this way. Evey single die roll has 216 combinations when combined with the other dice. 6 x 216 = 1296

Since 7 is the very highest number there is, each of the two 7's (both blue) have 216 combo's that they win at, giving you 432 total.

The simple solution is to think that Yellow will always roll a 4, so it's just a matter of whether the other people will beat it. If blue rolls a 7 then that wins, otherwise if Green rolls a 6 that wins, otherwise if Orange rolls a 5 that wins, otherwise Yellow does.
P(Blue) = 1/3
P(Green) = P(not Blue)*1/3 = 2/3 * 1/3 = 2/9
P(Orange) = P(not Blue)*P(not Green)*1/2 = 2/3*2/3 * 1/2 = 2/9
P(Yellow) = P(not Blue)*P(not Green)*P(not Orange) = 2/3*2/3*1/2 = 2/9

Interestingly if the question was you pick one die to play against an opponent who then makes their choice: you should pick Yellow or Orange, and your opponent picks the other. Any other choice means your opponent picks Yellow and has 2/3 chance of winning. (There are some sets of numbers such as 552222 444411 333333, where the first to pick is at a disadvantage - this is left to the reader to work it out!)