Discrete structures mathematics
two different factories both produce a certain automobile part. the probability that a component from the first factory is defective is 4%, and the probability that a component from the second factory is defective is 5%. in a supply of 160 of the parts, 100 were obtained from the first factory and 60 from the second factory.
(a) what is the probability that a part chosen at random from the 160 is from the first factory?
(b) what is the probability that a part chosen at random from the 160 is from the second factory?
(c) what is the probability that a part chosen at random from the 160 is defective?
(d) if the chosen part is defective, what is the probability that it came from the first factory?

a) 62.5%

b) 37.5%

c) 4.3%

d) 71%

Step-by-step explanation:

A=1: component from the 1st factory

A=2: component from the 2nd factory

X=0: component OK

X=1: component defective

a) P(A=1)=100/160=0.625=62.5%

b) P(A=2)=60/160=0.375=37.5%

c) P(X=1) = P(X=1 & A=1) + P(X=1 & A=2) = P(X=1|A=1).P(A=1) + P(X=1|A=2).P(A=2) = 0.04*100/160 + 0.05*60/160 = 0.025 + 0.1875 = 0.04375 = 4.3%

d) Based on Bayes’ theorem:

P(A=1|X=1) = P(X=1|A=1).P(A=1)/P(X=1) = 0.05*(100/160)/0.04375 = 5/7 = 0.71 = 71%

24: 56

step-by-step explanation:

24 can be divided into 3 and 56 can be divided into 7, where as all the other ratios can't.

solution:

→c= a b , that is product of a and b is possible only when when a is m × n matrix and b is n × p matrix.

that is, number of columns of a= number of rows of b.

so, size of matrix , c= a b is , if a = (m × n)matrix

b= (n × p)matrix

then a b = [ (m × n)matrix] ×[   (n × p)matrix ]

→→c= a b= [m × p ] matrix

→→size of product a b will be equal to = number of rows of a × number of columns of b