An elementarty school is offering 3 language classes: one in Spanish, one in Fre

Question

An elementarty school is offering 3 language classes: one in Spanish, one in French, and one in German. Theseclasses are open to any of the 100 students in the school. There are 28 students in Spanish class, 26 in French class, and 16in German class. There are 12 students that are in bothSpanish and French, 4 that are in both Spanish and German, and 6taht are in both French and German. In addition, there are 2students taking all 3 classes. a) If a student is chosen randomly, what is the probabilitythat he or she is not in any of these classes? b) If a student is chosen randomly, what is the probabilitythat he or she is taking exactly one language class? c) If 2 students are chosen randomly, what is the probabilitythat at least 1 is taking a language class? An elementarty school is offering 3 language classes: one in Spanish, one in French, and one in German. Theseclasses are open to any of the 100 students in the school. There are 28 students in Spanish class, 26 in French class, and 16in German class. There are 12 students that are in bothSpanish and French, 4 that are in both Spanish and German, and 6taht are in both French and German. In addition, there are 2students taking all 3 classes. a) If a student is chosen randomly, what is the probabilitythat he or she is not in any of these classes? b) If a student is chosen randomly, what is the probabilitythat he or she is taking exactly one language class? c) If 2 students are chosen randomly, what is the probabilitythat at least 1 is taking a language class?

Explanation / Answer

a) P(SUFUG) = P(S)+P(F)+P(G)-P(SF)-P(GF)-P(SG) + P(SFG)
= (28+26+16-12-6-4+2)/100 = 50/100 = 0.5
therefore ppl who did not take any of the course = 1- P(SUFUG)= 1-0.50 = 0.50
b) P(exactly one language) =  P(SUFUG)- P(SF) -P(GF)-P(SG) + 2*P(SFG) = 0.5 -(12+6+4)/100 +2*2/100
= 0.32... (the 2*P(SFG) term isbecause P(SF),P(GF),P(SG) alreadyincludes P(SFG) in each of them. So we aresubtracting this term thrice in stead of once. this is a correctionfor that. )
c) P( at least one is taking a language course) = 1- P(none istaking a language course)
= 1- P(1st one did not take any language course)*P(1st one didnot take any language course) = 1- 0.5*0.5 = 0.75
Hope it helps you. Feel free to ask for anyclarification.

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