Unintuitive difference in probabilities - $P(\text{Sum} \ge 10 | \text{First Dice} = 6)$ vs. $P(\text{Sum} \ge 10 | \text{One Dice} = 6)$

Question

Say I roll two die and record the sum. The outcomes would be:

I was finding two probabilities, and understand them individually, but I was quite confused behind why intuitively, the two are different:

The probability the first dice the sum is greater than or equal to $10$ given the first dice roll was a 6. $$P(\text{Sum} \ge 10 | \text{First Dice} = 6)=\frac{1}{2}$$ The probability the sum is greater than or equal to $10$ given one of the dice rolls was a 6. $$P(\text{Sum} \ge 10 | \text{One of the Dice} = 6) = \frac{5}{11}$$

I get how they're calculated, but what is the intuition behind why the second probability is less than the first? - It looks symmetric to me; I would have expected them to be the same!

Answer 1

The second conditional probability should read ANY one dice is 6. The intuition is simply that it doesn't allow the outcome (6,6) while the first probability does.

Answer 2

the domain restricted to the first case is the following in blue

the domain restricted to the second case is the following in red

The solution, intuitive or not is self evident

Answer 3

If you're given "one of the dice is $6$" (which does mean not exactly one of them but at least one of them), you rule out fewer options (namely $25$) than with saying "die 1 is $6$" which rules out $30$ possible outcomes. So you know less in one case.