Roll a dice 3 times. What is the probability to get 6 at least once?

Question

There are two answers to this problem:

1. First roll, second roll, and third roll are mutually exclusive events. Hence, $$P(A) = 3 * \frac{1}{6} = 50\%$$
2. These three events are not mutually exclusive. Hence, $$P(A) = 1 - (\frac{5}{6})^3 = 42\%$$

I can not convince myself why 3 independent rolls are not mutually exclusive. I don't care do 1 get 1, 2, or 3 6's. What kind of argument would you use to convince others that (2) is correct?

Answer 1

The events "first roll is a $6$" and "second roll is a $6$" are not mutually exclusive, since both events can occur at the same time ("first two rolls are both sixes").

Answer 2

As other responses indicated, you have (in general) confused:

two events are mutually exclusive

and two events are independent

Let A and B be two different events.

Let the probability of each event happening be $p(A)$ and $p(B)$, respectively.

If $A$ and $B$ are mutually exclusive events, then it is impossible for both events to occur.

If $A$ and $B$ are independent events, then the chance of event $A$ occurring is totally unaffected by whether or not event $B$ also occurs.

Mathematically, this 2nd property is expressed as
$p(A) = p(A|B) = p(A ~\text{given that event B occurred}).$

Let (for example)
event A denote getting a 6 on the 1st roll
event B denote getting a 6 on the 2nd roll

Is it impossible for both events to occur?
Will the chances of (for example) event B occurring be affected by whether event A occurred?