A school has 100 students The school offers only 3 language

A school has 100 students. The school offers only 3 language classes, namely Spanish, Italian and Russian. 50 students do not take a language. The Spanish, Italian, and Russian classes have 28, 26, and 16 respectively. However, 12 students take both Spanish and Italian, 4 students take both Spanish and Russian, and 6 students take both Italian and Russian.

1. How many students take all 3 language classes?

2. What is the probability a randomly chosen student takes exactly 1 class?

3. You randomly draw 2 students. What is the probability that they are taking at least 1 language class between them?

Solution

P(S) = 28;
P(I) = 26
P(R) = 16
P( S n I) = 12
P( S n R) = 4
P( I n R) = 6
P( do n\'t take language) = 50

a)
P(S U I U R) = P(S) + P(I) + P(R) + P( S n I) + P( S n R) + P( I n R) + P( S n I n R)
50 = 28 + 26 + 16 - 12 - 4 - 6 + P( S n I n R)
P( S n I n R) = 50 -28 -26 -16 + 12 + 4 + 6 = 2

b)
P( Exactly one class) = P( Spanish only) + P( Italian Only) + P(Russian Onnly) = [ P(S) - P( S n I) - P( S n R ) + P( S n I n R) ] + [ P(I) - P( S n I) - P( I n R) + P( S n I n R) ] + [ P(R) - P( S n R) - P( I n R) + P( S n I n R) ]
       = [ 28 - 12 - 4 + 3 ] + [ 26 - 4 - 6 + 2 ] + [ 16 - 12 - 6 + 2 ]/ 100 = 33/ 100 = 0.33