Jane Street Interview Question: 2 fair dice. What is the prob... | Glassdoor.co.uk

# 2 fair dice. What is the probability of both showing six if

I have observed at least one six.

2

1/6. Independent probabilities, so seeing one makes no difference to the second.

Anonymous on 31 Jul 2012
5

Not quite... The subtlety is "at least one". From two dice rolls there are 6*6 = 36 distinct outcomes, of these, 11 contain at least one six, i.e. (6,1), (6,2), (6,3), (6,4), (6,5), (6,6), (5,6), (4,6), (3,6), (2,6), (1,6), therefore P( (6,6) | at least one 6 ) = 1 / 11.

njm on 1 Aug 2012
5

More formally:
P(observe one 6 ) = P( 1st die is 6 OR 2nd die is 6 ) = P(1st is 6) + P(2nd is 6) - P(1st 6)*P(2nd 6) = 1/6 + 1/6 - 1/36 = 11/36, by reunion.
Now, we want to calculate P(both are 6's | observed one 6 ) = P ( observe one 6 | both are 6's) * P ( both are 6's) / P ( observe one 6 ) = 100% * 1/36 * 36/11 , where last factor was calculated above. So, all together P (both 6's| obs. one 6) = 1/11

1

11 possibilities of getting at least one six (6 x 1 + 1 x 6 - 1)
so P = 1/11

Me on 3 May 2014
1

I disagree with the above:

P(rolling at least one 6) = 1 - p(rolling no 6).

rolling no 6 twice in a row = (5/6)^2.

= 25/36

therefore 1 - 25/36 = 11/36. I thought njm had answered it halfway through his answer but then somehow arrived at 1/11? I believe he looked at 11 outcomes, instead of the initial 36.

WallStreetWolf on 3 Sep 2014
0

Ah I do apologise, mis read the question! yes you are all right! ha

WallStreetWolf on 3 Sep 2014