Why are the two events independent of each other in this case?

Question

For the following problem, the two events are said to be independent of each other.

Intuitively, I would think that the outcome of 10 depends on getting 5 on both rolls, which gives a YES answer. Can someone explain this to me?

Answer 1

You simply mixed up the linguistics of the question, by ignoring a negation. Inititally you say

For the following problem, the two events are said to be independent of each other.

That is not what is indicated in the picture. The question asks if the events are $\color{red}{\rm in}\rm dependent$, and "No" is presented as correct. That's the same as saying they are dependent.

Which is what your intuition told you and which paw88789 proved in their answer.

Your inituition and the answer (from an online test, I presume) agree. They are just worded differently ("dependent" vs. "not independent").

Answer 2

Letting $A$ be the sum is $10$ (as in OP's statement), and letting $B$ be the event of at least one $5$, then:

$P(A)=\frac{5}{25}$

$P(B)=\frac{9}{25}$

and $P(A\cap B)=P(A)=\frac{1}{25}$.

Since $P(A\cap B)\ne P(A)\cdot P(B)$, the events are not independent.

One could also reason that event $A$ is a subset of event $B$. So $P(B|A)=1\ne P(B)$