If p(a) = 2/3, p(b) = 4/5, and p(image attached)

a. 11/15
b. 13/25.
c. 8/15
d. 14/15

P(A ∩ B) = 11/15 ⇒ answer A

Step-by-step explanation:

* Lets revise the meaning of ∪ and ∩

# A ∪ B means all the elements in A or B without reputation

- Ex: If A = {2 , 3 , 5} and B = {3 , 4 , 7}

∴ A ∪ B = {2 , 3 , 4 , 5 , 7} ⇒ we don't repeat the element 3

# A ∩ B means all the elements in A and B

- Ex: If A = {2 , 3 , 5} and B = {3 , 4 , 7}

∴ A ∩ B = {3}

- From the examples above

∵ n(A) = 3 and n(B) = 3

∵ n(A ∪ B) = 5

∵ n(A ∩ B) = 1

∴ n(A) + n(B) = n(A ∪ B) + n(A ∩ B)

* Now lets solve the problem

∵ P(A ∪ B) = 11/15

∵ P(x) = n(x)/total

- That means the total elements in the problem is 15 and n(A ∪ B) is 11

∴ n(A ∪ B) = 11

∵ P(A) = 2/3 ⇒ simplest form

- To find P(A) without simplification and you now the total is 15

  then multiply up and down by 5

∴ P(A) = (2×5)/(3×5) = 10/15

∴ n(A) = 10

∵ P(B) = 4/5 ⇒ simplest form

- To find P(B) without simplification and you now the total is 15

  then multiply up and down by 3

∴ P(B) = (4×3)/(5×3) = 12/15

∴ n(B) = 12

- To find n(A ∩ B) use the rule above

∵ n(A) + n(B) = n(A ∪ B) + n(A ∩ B)

∵ 10 + 12 = 11 + n(A ∩ B) ⇒ subtract 11 from both sides

∴ 11 = n(A ∩ B)

- The number of elements in A ∩ B is 11

∵ P(A ∩ B) = n(A ∩ B)/total

∴ P(A ∩ B) = 11/15