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how do i find the number of elements in this set??

I have this question regarding set theory and probability that i am struggling on:

Q: A group of international students take part in a survey on the nationality of their parents.
E= English Parent
F= French parent

n(ξ) = 50,  n(E) = 15,  n(F )= 9,  n(E U F)' = 33

     (1) Find  n(E ∩ F)                          

     (2) Find n(E' U F)

     (3) A student is chosen at random. Find the probability that this student has an        English parent and a French parent.

    (4) A student who has a French parent is chosen at random. Find the probability that this student also has an English parent.


Thankyou so much for any help ! 

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1 Answer

First thing to do is draw a Venn diagram. It will help tremendously.

If n(E U F)' = 33, then n(E U F) = 50 - 33 = 17, because n(E U F) + n(E U F)' = n(total).

We also know that n(E U F) = n(E) + n(F) - n(E ∩ F). Substitute values to get 17 = 15 + 9 - n(E ∩ F). Solve for n(E ∩ F) and get n(E ∩ F) = 7.

n(E' U F) is the number of students without an English parent plus the number of students with a French parent, which is the total number of students minus the number with an English parent, plus the number with a French parent: n(total) - n(E) + n(F) = 50 - 15 + 9 = 44.

Probability of a student with an E and and F is the probability of the union of E and F.

For the last one, it will be the intersection of E and F. I highly recommend you draw a Venn diagram and work this through!