Probability Strategy Guide Manhattan Q#1 and PG 65

[VikT37]

What is the probability that the sum of two dices will yield a 10 or lower.
This is manhattan strategy guide question. (Q #1 and Pg : 65)

the solution discusses takes all the possibilities and comes up with Probability : 11/12.

I fail to understand where is my logic wrong. I took all the possible sum of two fair dices : 2,3,4,5,6,7,8,9,10,11,12
we need sum 10 or lower : which is : 2,3,4,5,6,7,8,9,10

Total outcomes of sum : 11
Favorable outcomes : 9
therefore probability is : 9/11 ( which is wrong) I fail to understand the gap in my logic. Why can't we get answer this way.
thanks!!!

[tim]

You're not supposed to take all possible SUMS, because many of those sums can come about in multiple ways. (This is why you get better odds for a 4 on a craps table then a 6!) What you need to do is take all possible ROLLS:

1-1
1-2
1-3
1-4
1-5
1-6
2-1
2-2
etc.

Of these 36 rolls (6 choices for the first die times 6 choices for the second die), it's easiest to figure out how many DON'T give you a 10 or less, i.e. they give you 11 or 12. Those rolls are:

5-6
6-5
6-6

3 of the 36 rolls give you more than 10, so the other 33 give you 10 or less. 33/36 = 11/12

[RonPurewal]

I fail to understand where is my logic wrong. I took all the possible sum of two fair dices : 2,3,4,5,6,7,8,9,10,11,12

tim's method is a mathematically correct approach, but it doesn't explicitly address why YOUR approach is not correct.

your approach is incorrect because these are not equally likely. if you are going to calculate probability by counting possibilities, then those possibilities MUST be equally likely.

the different sums are not equally likely; some of them are vastly more likely than others. (e.g., there are six different ways to roll a sum of 7, but only one way to roll a sum of 2 or 12.)

[RonPurewal]

analogy:
let's say i play the lottery.

technically, i could say that there are just two outcomes: 'i win' and 'i don't win'.
by the logic above, then, i would have a 50 per cent chance of winning the lottery.
woohoo!
(too bad it doesn't actually work that way...)