Dice question?

Question

There are 4 dice and 2 players are involved in a game. The first player selects 1 of the 4 dice and the second player selects 1 of the next 3. Each player rolls his die and the higher # wins.

The first player selects a die with four 4's & two 0's...he wins 30 tokens every time he wins. Which of the next 3 dice should the second player choose?
-die with all 3's
-die with four 2's & two 6's
-die with three 5's & 3 1's

Why?

Answer 1

This problem is actually quite simple, you have to figure out the probability of winning, based on the throws of the dice. Because the two events are independent then you multiply the probabilities to find out the chance that they will happen together.

If you choose the die with 4 2’s and 2 6’s:
You will lose when you roll a 2 and they roll a 4, that will happen (4/6)*(4/6) or 16/36 times
You will win when you roll a 2 and they roll a 0, that will happen (4/6)*(2/6) or 8/36 times
You will win when you roll a 6 and they roll a 0, that will happen (2/6)*(2/6) or 4/36 times
You will win when you roll a 6 and they roll a 4, that will happen (2/6)*(4/6) or 8/36 times

So with the die with 4 2’s and 2 6’s you will win 20/36 rolls and lose 16/36 rolls

If you choose the die with 6 3’s:
You will lose when you roll a 3 and they roll a 4, that will happen (6/6)*(4/6) or 24/36 times
You will win when you roll a 3 and they roll a 0, that will happen (6/6)*(2/6) or 12/36 times

So with the die with 6 3’s you will win 12/36 rolls and lose 24/36 rolls

If you choose the die with 3 5’s and 3 1’s:
You will lose when you roll a 1 and they roll a 4, that will happen (3/6)*(4/6) or 12/36 times
You will win when you roll a 1 and they roll a 0, that will happen (3/6)*(2/6) or 6/36 times
You will win when you roll a 5 and they roll a 0, that will happen (3/6)*(2/6) or 6/36 times
You will win when you roll a 5 and they roll a 4, that will happen (3/6)*(4/6) or 12/36 times

So with the die with 4 2’s and 2 6’s you will win 12/36 rolls and lose 24/36 rolls

The obvious choice is the die with 3 5’s and 3 1’s

Answer 2

Even if it's long, it's worth it:
He should choose the dice with better probability to win against his friend. All 3's has a 2/6 possible chance of winning since it only beats his friends dice 2 times out of all possible outcomes. 4 twos and 2 sixes has 2/6 for sure plus a 2/6 chance of possibly winning since it wins every time with 6's but can only win 2/6 times with 2's. this dice is so far better since it has a chance of winning for sure if it rolls as six where as with all 3's it depends what the other person rolls. 3 fives and 3 ones will win half of the time or has a for sure 3/6 chance. It also has a possible 3/6 win. Since dice 2 was best with a 2/6 for sure chance and 2/6 slight chance, dice three beats it by one on each since it has 3 out of the six sides to win by for sure and 3/6 slight chance depending on what they roll. For sure chance=if you roll the die, if you get a certain result you atomatically win.
slight chance = if you roll the die, depending on what they get is the outcome. You do not compare the slight chance and for sure chance for a dice.

Dice 3 is best, then dice 2 , then dice 1 is worst, so he should choose 3 fives and 3 ones if he wants to have the best chance to win.

Answer 3

Your opponent will roll a 4 2/3 of the time.
You can win at least 1/2 of the time with the die that has the three 5s.
That's the one to pick

Answer 4

die with three 5's & 3 1's

all dice give guarenteed to beat the 0s
this dice has 3 chances to beat the fours as opposed to 2 chances and 0 chances on the other dice