Which of the following statements is correct given that the events a and b have nonzero probabilities? a. a and b can be both independent and mutually exclusive b. a and b cannot be both independent and mutually exclusive c. a and b are always independent d. a and b are always mutually exclusive

The statement which is correct is:

B.     A and B cannot be both independent and mutually exclusive.

Let we toss a coin single time.

and event A denote the tossing up a head.

and event B denote tossing a tail.

                               P(A∩B)=0

                         P(A∩B)=P(A)×P(B)

Here we have:

P(A)≠0 and P(B)≠0

So, if the events are independent then we have:

P(A∩B)≠0

Hence, the events can't be mutually exclusive.

Similarly if the two events are mutually exclusive then they can't be independent.

i.e. both mutually exclusive property and independent property can't exist at the same time if the two events have non-zero probability.

     Hence, the answer is: Option: B

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1