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Advanced Math/Rolling One Dice Twice - Probability
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Question
Hello,
Here we are on a snowy evening in the UK playing Yahtzee.
Please can you explain the probability of shaking a particular number with one dice twice, as opposed to two dice once.
For example, I have already shaken 4 sixes on the other 4 dice, so to get a Yahtzee I need to shake a six with my last remaining dice.
The probability of shaking a six in the first shake is 1/6 and it would appear that the probability of shaking a six in the second shake is 1/6.
So is the probability of shaking a six in one or other of the two remaining goes 2/12 ie.1/6?
I understand the probability calculations for shaking a six with two dice and realise that they are different.
I look forward to receiving your comments and thank you in advance!
Belinda
The probability function for one die is
# p
1 1/6
2 1/6
3 1/6
4 1/6
5 1/6
6 1/6
To get the probablity function for two dice, note that since each number has probability 1/6,
that would mean the second roll would have probability 1/6 of 1/6, or 1/36.
So if the first die was a 1, the second die would give the sum
# p
2 1/36
3 1/36
4 1/36
5 1/36
6 1/36
7 1/36
Taking this into account for 6 possibilities on the first die, we get the same table
starting with a 3, 4, 5, 6, and 7.
This is equivalant to 1/36 chance of a 2, 2/36 chance of a 3 on up to 6/36 chance of a 7,
then 5/36 chance of an 8 down to 1/36 chance of a 12. This looks like
# p
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36.
As far as the probability of getting a 6 on either die, the rolls would be
1,6 / 2,6 / 3,6 / 4,6 / 5,6 / 6,6 / 6,5 / 6,4 / 6,3 / 6,2 / 6,1 - 11 ways to do it.
Let A and be a 6 on the first die and B be a 6 on the second die.
This can be found by P(AuB) = P(A) + P(B) - P(AB)
It is known that P(A) = 1/6 and P(B) = 1/6, and that the chance of them both being a 6 is the product of the two, so P(AB)=1/36.
The chances of a 6 are then 1/6 + 1/6 - 1/36 = 6/36 + 6/36 - 1/36 = 11/36.
Note that this is a little less than 1/3.
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