-aux.02 Venn diagram sample space U and events A and B

[karush]

(a) the shaded hopefully shows $(A\cup B)'$

(b) (i) if $(A\cup B)'= 21$ and $n(U)=36$ then $n(A\cup B)=15$
but $n(A)+n(B)=17$ so $n(A\cap B) = 2$

(ii) $P(A\cap B)$ not sure but guessing $2:17$

(c) not sure what "mutually exclusive" means but presume it has to do with the overlap.

never done Venn Diagrams so this is all new ... did look at De Morgan's stuff tho..

 

[MarkFL]

(a) Correct. Anything that is not in either A or B should be shaded, which is what you did.

(b)

(i) Correct.

(ii) I believe you want to use:

\(\displaystyle P(A\cap B)=\frac{n(A\cap B)}{n(U)}\)

(c) Two sets are mutually exclusive if:

\(\displaystyle A\cap B=\emptyset\)

 

[karush]

(a)

(ii) I believe you want to use:

\(\displaystyle P(A\cap B)=\frac{n(A\cap B)}{n(U)}\)

so

$\displaystyle P(A\cap B)=\frac{n(A\cap B)}{n(U)}=

\frac{2}{36}=\frac{1}{16}$

 

[MarkFL]

so

$\displaystyle P(A\cap B)=\frac{n(A\cap B)}{n(U)}=

\frac{2}{36}=\frac{1}{16}$

Not quite...

\(\displaystyle \frac{2}{36}=\frac{2}{2\cdot18}=\frac{1}{18}\)

:D

 

[CellCoree]

I can provide a more mathematical explanation to the content provided. In a Venn diagram, the shaded region represents the complement of the union of events A and B, denoted as $(A\cup B)'$. This means that the shaded region includes all elements that are not in either A or B.

In part (b), it is given that the size of the sample space U is 36 and the size of the complement of the union of A and B is 21. From this, we can calculate the size of the union of A and B as 36-21 = 15. However, we also know that the size of A and B individually is 17. This means that the size of the intersection of A and B must be 17-15 = 2.

For part (ii), we are asked to calculate the probability of the intersection of A and B, denoted as $P(A\cap B)$. Since we know the size of the intersection is 2 and the size of the sample space is 36, the probability can be calculated as 2/36 = 1/18.

The term "mutually exclusive" means that two events cannot occur at the same time. In a Venn diagram, this means that the two events A and B have no overlap, i.e. their intersection is empty. However, in this case, we have found that the size of the intersection is 2, which means that A and B are not mutually exclusive. This is also supported by the fact that the size of the union of A and B is less than the sum of their individual sizes.