Question

5.) Assume 365 days in the year and that people's birthdays are randomly distributed throughout the year. With 16 people in the room, what is the probability that at least 2 have the same birthday? How many people are required so that the probability is at least 50%?

6.) A factory has 3 machines (A, B,&C) A makes 20% of the parts produced by the factory, B makes 30% of the parts and C makes 50% of the parts. 6% of the parts made by A are defective, 7% of the parts made by B are defective and 8% of the parts made by C are defective. Al the parts are tossed in a single box. What is the probability that a part picked from the box is defective? Suppose that a part is selected from the box at day's end and is found to be defective. What is the probability that C made it? That B made it? That A made it?

Answer 1

5)

P( at least 2 have the same birthday) =P(all 16 have different birthdays) =1-(₃₆₅P₁₆)/365¹⁶ =1-0.7164

=0.2836

let number of people required =a

therefore P( at least 2 have the same birthday) =P(all n have different birthdays) =1--(₃₆₅P_n)/365ⁿ >=0.5

solving for n by putting values, minimum number required so that the probability is at least 50% =23

6)

P(defective )=P(A)*P(defective|A)+P(B)*P(defective|B)+P(C)*P(defective|C)

=0.2*0.06+0.3*0.07+0.5*0.08 =0.073

P(A|defective) =P(A)*P(defective|A)/P(defective) ==0.2*0.06/0.073 =0.1644

P(B|defective) =P(B)*P(defective|B)/P(defective) ==0.3*0.07/0.073 =0.2877

P(C|defective) =P(C)*P(defective|C)/P(defective) ==0.5*0.08/0.073 =0.5479