4. (15 points total) (a) (5 points) A box contains 5 marbles, 3 Red, 1 Green, 1

Question

4. (15 points total) (a) (5 points) A box contains 5 marbles, 3 Red, 1 Green, 1 Blue. Consider an experiment that consists of taking 1 marble from the box, then replacing it in the box and drawing a second marble from the box. Describe the sample space. repeat when the second marble is drawn without first replacing the first marble. (b) (5 points) What is the probability that the first marble was Green and the second was Blue with replacement? (c) (5 points) What is the probability that the first marble was Green and the second was Blue without replacement? 5. (15 points total) An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. These classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 in the French class and 26 in the German class. There are 12 students that are in both Spanish and French, 5 that are in both Spanish and German, and 6 that are in both French and German. In addition, there are 3 students taking all 3 classes. (a) (5 points) If a student is chosen randomly, what is the probability that they are not in any of these classes. (b) (5 points) If a student is chosen randomly, what is the probability that they are taking exactly one language class. (c) (5 points) If 2 students are chosen randomly, what is the probability that at least one is taking a language class.

Explanation / Answer

4)

a)

here let drawing Red ; Green and Blue marble is event R ; G and B

therefore

with replacement sample space S ={RR,GG, BB,RG,GR,RB,BR,BG,GB}

without replacement : sample space S ={RR,RG,GR,RB,BR,BG,GB}

b)

P(first green and second blue) =(1/5)*(1/5) =1/25 (as there are only 1 green and 1 blue marble and sample is taken with replacement therefore total marble will remain 5 in both cases)

c)P(first green and second blue) =(1/5)*(1/4) =1/20

5)

let N is total number of people ; A,B and C are student learning Spanish; French and German

a)

therefore number who are not in any class =100-60 =40

P(randomly chosen not in any class)=40/100=0.4

b)

P(taking exactly one clas)=43/100=0.43

c)

P(at least one is taking language class)=1-P(none is taking) =1-(40/100)*(39/99)=0.8424

N(T)= 100 N(A)= 28 N(B)= 26 N(C)= 26 N(AnB)= 12 N(BnC)= 6 N(AnC)= 5

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