Question

5) A bin contains 62 bolts, 28 have metric thread while the remainder have English thread. Suppose that 3 washers are selected without replacement from the lot. A) What is the probability that all 3 bolts have metric thread? B) What is the probability that the third bolt selected has English thread if the first and second bolts selected have metric threads? C) What is the probability that the second bolt selected has English thread if the first and third bolts selected have metric threads? 6) Transmissions for a given model of car are built in 3 different factories. It is determined that: Factory A produces 32% of the transmissions used, and 1 in 178 are defective. Factory B produces 4 1% of the transmissions used, and 1 in 220 are defective. Factory C produces 27% of the transmissions used, and 1 in 204 are defective. . If a randomly selected car has a defective transmission, what is the probability the transmission was built in factory B? 喇Consider the following two step experiment. The first step is to toss a coin and record the outcome. The second step is to toss the coin again and record the outcome. A) Construct the sample space (the set of all possible outcomes) for this experiment when order matters. B) Construct the sample space (the set of all possible outcomes) for this experiment when order does not matter. Consider a random process involving the repeated tossing of a coin. The process begins with the first toss of the coin and ends when either of the following conditions occur. .A tail occurs on the toss after a head occurred. . A total of 4 tosses has occurred. A) Construct the sample space for this random process. B) Does order matter for a given outcome of this random process?

Answer 1

$5.~A.~We~can~select~3~bolts~in~\binom{62}{3}~ways~which~are~all~possible~equally\\ likely~cases.~Let~A~be~an~event~that~all~3~bolts~have~metric~thread,~then\\ total~no.~of~cases~favourable~to~event~A=\binom{28}{3}. \\Hence~P(A)=\frac{\binom{28}{3}}{\binom{62}{3}}=0.0866\\ B.~Let~A~be~an~event~that~third~bolt~selected~has~English~thread\\ B~be~event~that~the~first~and~ ~second~selected~have~metric~threads\\ P(A\cap B)=\frac{28*27*34}{62*61*60},~P(B)=\frac{28*27}{62*61}\\ Required~probability,~P(A|B)=\frac{34}{60}=0.5667\\ C.~~Let~A~be~an~event~that~second~bolt~selected~has~English~thread\\ B~be~event~that~the~first~and~ ~third~selected~have~metric~threads\\ P(A\cap B)=\frac{28*34*27}{62*61*60},~P(B)=\frac{28*61*27}{62*61*60} \\Required~Probability=P(A|B)=\frac{34}{61}=0.5667$

$6.~Let~D~be~an~event~of~defective~transmission\\ Let~A,~B,~C~be~events~of~producing~transmission~from~Factories~A,B,C~respectively.\\ According~to~the~problem,~P(A)=0.32,~P(D|A)=1/178,~P(B)=0.41,\\ ~P(D|B)=1/220,~P(C)=0.27,~P(D|C)=1/204\\ Required~probability=P(B|D)=\frac{P(D|B)P(B)}{P(D|A)P(A)+P(D|B)P(B)+P(D|C)P(C)} \\ =(0.41*(1/220))/(0.32*(1/178)+0.41*(1/220)+0.27*(1/204))\\ =0.3739\\ A)~Sample~space=\{(H,H),(H,T),(T,H),(T,T)\},~where~H=head,\\~T=tail\\ B)~Sample~space=\{(H,H),(H,T),(T,T)\}\\ 8.~A)~Sample~space=\{HT,~HHT,~HHHT,~HHHH,~THT,THHT,THHH,\\ TTHH,TTHT,TTTH,TTTT\}\\ B)~Yes.$