Question on fraction probability

Question

I was doing ok until they threw fractions at me lol.  Here’s the question.

 

One third of children in class like hot dogs, 1/2 like pizza, and 3/4 like hot dogs or pizza.  What is the probability that a randomly selected child likes both hot dogs and pizza?

Answer 1

draw a Venn diagram with circle "HD" partiaqlly overlapping circle "P".

Within this figure-8 diagram, place labels a, b and c into the three distinct areas, from left to right;

"a" represents fraction that like hot dogs only; "b" represents fraction that like both; "c" represents fraction that like pizza only.

 

The given statements tell us that a+b = 1/3,  c+b = 1/2, and  a+b+c = 3/4.

 

You can solve this system because it has three unknowns and there are three distinct equations.

 

I will not spoil the fun of solving this system for you.  

Answer 2

Remember: math problems are more afraid of you than you are of them!

The probability formula for simultaneous events is:

P(A or B) = P(A) + P(B) - P(A & B)

In your case, we rewrite it as:

(P(pizza or hot dogs) = P(pizza) + P(hot dogs) - P(pizza & hot dogs)

3/4       =   1/2         +     1/3       - P(both)

3/4       =               5/6                  - P(both)

-5/6                       -5/6

-1/12    =                           - P(both)

/ -1                                      / -1

1/12     =                    P (both)

The formula subtracts the "AND" because we counted things twice; some of our pizza lovers also liked hot dogs and vice versus. Removing the double-count keeps us honest.