Let's start with the first one.
The probability of rolling the same die in each die is the numer of possible ways we can do that divided by the total possible outcomes.
Since each die has 6 outcomes, rolling both will have 6² = 36 outcomes.
1 - Now, there is only 6 possible outcomes in which we roll the same digit in each die:
1 -- 1
2 -- 2
3 -- 3
4 -- 4
5 -- 5
6 -- 6
Thus, the probability of this happaning is:
This is not 1/4, so the first statement is false.
2 - To roll a 10, we have to add both to make 10. the ways of doing that are:
4 + 6 = 10
5 + 5 = 10
6 + 4 = 10
There is no other way, because any roll 3 or lower won't be able to reach a sum of 10. Thus there is 3 possibilities, but no we want the odds, that is, the number of outcomes of the result we want over the number of outcomes of the result not happening. If there is 3 possible ways it happens and 36 outcomes in total, there is 36 - 3 = 33 outcomes in which we won't roll a 10. So, the odds in favor of rolling a 10 is:
This is 1:11, not 1:13, so the second statement is false.
3 - Let's start with the outcomes of 2 (we can't get one, becase the lowest is 1 + 1 = 2). There is only one outcome for 2:
1 + 1 = 2
For 3, we have two:
1 + 2 = 3
2 + 1 = 3
For 4, we have three:
1 + 3 = 4
2 + 2 = 5
3 + 1 = 4
For 5, we have four:
1 + 4 = 5
2 + 3 = 5
3 + 2 = 5
4 + 1 = 5
And for 6, we have five:
1 + 5 = 6
2 + 4 = 6
3 + 3 = 6
4 + 2 = 6
5 + 1 = 6
So, in total, the outcomes for rolling less than 7 is
Since the total outcomes is 36, there are 36 - 15 = 21.
The odds against rolling a number less than 7 is the number of outcomes of this doesn't happening over the number of outcomes for when this happens, so:
That is, the odds are 7:5, which matches the statement, so the third statement if true.
Answers:
1 - false
2 - false
3 - true