Probability Question - Please Help

Question

Two diseases A and B have spread within a certain population. We
know that 15% of this population have the disease A, 20% of this
population have disease B and 25% have at least one of the two diseases.
We choose a random person among this population.

(a) What is the probability that this person contracted both diseases?
(b) Given that the chosen individual has at least one of the diseases,
what is the probability that this person has contracted both dis-
eases?

Answer 1

Start by making a table like so.

             B:Yes         B:No

A:Yes       W              X

A:No         Y               Z

 

We know that 15% have disease A, so 1: W+X = 15%.

We know that 20% have disease B, so 2: W+Y = 20%.

We know that 25% have at least one of the two diseases, so 3: W+X+Y = 25%.

The four entries in the table total the entire population, so 4: W+X+Y+Z = 100%

Subtracting equation 2 from equation 3 gives us X = 5%.

Subtracting equation 1 from equation 3 gives us Y = 10%.

Plugging these in to equation 3 gives W + 5% + 10%= 25%, so W = 10%.

Finally, subtracting equation 3 from equation 4 gives us Z = 75%.

We can now fill in the table:

               B:Yes         B:No

A:Yes        10%          5%

A:No         10%          75%

You should now be able to answer both questions.