Here is the titanic data we are working with... 

         died | survived | totals
female|   126 |    344   |  470
 male |  1364 |    366   | 1730
totals|  1490 |    710   | 2200 

We are asked to show why the general addition rule 
is true and the hint was to use Venn Diagrams.

Draw a rectangle on the left for all the Males
and label it M:

 

To the right of that, draw a rectangle for all the
Females and label it F:

 

Now draw a circle in the middle for the survivors
and label it S

 

The left half of the circle contains the male
survivors, so put 366 in the left half of the
circle:



The males who died are in the left rectangle
but outside the circle. So put 1364 there:



The right half of the circle contains the female
survivors, so put 344 in the right half of the
circle:



The females who died are in the right rectangle
but outside the circle. So put 126 there:



Now to get P(M or S), we just look at this
part of the Venn diagram:



So the number in this part is 1364+366+344 = 2074
and since there were 2200 people on board, the
probablity that a person on board was either male
or survived is given by

P(M or S) =  =  

Now let's use the formula and see if we get the same answer:

P(M or S) = P(M) + P(S) - P(M and S)

P(M or S) =  +  - 

P(M or S} =  =  = 

So we see that the general sum formula does agree 
with the Venn diagram answer.

We can also do the same thing with P(F or S)

Now to get P(F or S), we just look at this
part of the Venn diagram:



So the number in this part is 366+344+126 = 836
and since there were 2200 people on board, the
probablity that a person on board was either female
or survived is given by

P(F or S) =  =  

Now let's use the formula and see if we get the same answer:

P(F or S) = P(F) + P(S) - P(F and S)

P(F or S) =  +  - 

P(F or S} =  =  = 

So we see again that the general sum formula 
does agree with the Venn diagram answer.

Edwin