Given P(A) = .70, P(B) = .30, and P(A ∩ B) = .00, find (a) P(A ∪ B) and (b) P(A | B).

(c) Sketch a Venn diagram and describe it in words

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 1

I think you'll find that given two possible outcomes, it will always be one or the other. :-)

P(A|B) = P(A ∩ B)/P(B)
Since P(A and B) = 0, P(A|B) = 0

To find the probabilities requested, we'll use the formulas:

(a) P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
(b) P(A | B) = P(A ∩ B) / P(B)

Given:
P(A) = 0.70
P(B) = 0.30
P(A ∩ B) = 0.00

(a) To find P(A ∪ B), we substitute the given values in the formula:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) = 0.70 + 0.30 - 0.00
P(A ∪ B) = 1.00

(b) To find P(A | B), we substitute the given values in the formula:
P(A | B) = P(A ∩ B) / P(B)
P(A | B) = 0.00 / 0.30
P(A | B) = 0.00

(c) Sketching a Venn diagram:

A B

Since P(A ∩ B) = 0.00, it means that A and B have no intersection. In other words, they are mutually exclusive.

The Venn diagram would consist of two separate circles, one representing A and the other representing B. There is no overlapping area between the circles, indicating that the events A and B do not occur together. It visually represents the fact that P(A ∩ B) = 0.00.

To find the answers for (a) P(A ∪ B) and (b) P(A | B), we need to use the formulas related to probability.

(a) P(A ∪ B) represents the probability of either event A or event B occurring or both occurring. To find this probability, we can use the formula:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Given that P(A) = 0.70, P(B) = 0.30, and P(A ∩ B) = 0.00, we can substitute these values into the formula:

P(A ∪ B) = 0.70 + 0.30 - 0.00

Simplifying the equation, we get:

P(A ∪ B) = 1.00

Therefore, the probability of event A or event B occurring, or both occurring, is 1.00, which represents certainty.

(b) P(A | B) represents the probability of event A occurring given that event B has already occurred. The formula to calculate this conditional probability is:

P(A | B) = P(A ∩ B) / P(B)

However, in this case, P(A ∩ B) = 0.00, and P(B) = 0.30, which means we have a division by zero. Therefore, the probability P(A | B) is undefined since there is no overlap between events A and B.

(c) In a Venn diagram, the two circles represent event A and event B, respectively. Given that P(A) = 0.70 and P(B) = 0.30, we can draw a circle labeled "A" which encompasses 70% of the total area and a circle labeled "B" which encompasses 30% of the total area. Since P(A ∩ B) = 0.00, the circles do not overlap at all.

In words, the Venn diagram represents two separate events, A and B, with no common elements.