CS 70Discrete Mathematics and Probability TheoryHW 8Spring 2018Satish Rao and Babak AyazifarSundryBefore you start your homework, write down your team.
... [Show More] Who else did you work with on thishomework? List names and email addresses. (In case of homework party, you can also just describethe group.) How did you work on this homework? Working in groups of 3-5 will earn credit foryour "Sundry" grade.Please copy the following statement and sign next to it:I certify that all solutions are entirely in my words and that I have not looked at another student'ssolutions. I have credited all external sources in this write up.I certify that all solutions are entirely in my words and that I have not looked at another student'ssolutions. I have credited all external sources in this write up. (signature here)
1Faulty LightbulbsBox 1 contains 1000 lightbulbs of which 10% are defective. Box 2 contains 2000 lightbulbs ofwhich 5% are defective.(a) Suppose a box is given to you at random and you randomly select a lightbulb from the box. Ifthat lightbulb is defective, what is the probability you chose Box 1?(b) Suppose now that a box is given to you at random and you randomly select two light- bulbsfrom the box. If both lightbulbs are defective, what is the probability that you chose from Box1?Solution:(a) Let:•Ddenote the event that the lightbulb we selected is defective.•Bdenote the event that the lightbulb we selected is from Boxi.iWe wish to computeP[B|D]. Using Bayes' Rule we get:1
CS 70, Spring 2018, HW 81
P[D|B]·P[B]11P[B|D]=1P[B]·P[D|B]+P[B]·P[D|B]11220.1·0.5=0.5·0.1+0.5·0.052=3(b) Let:0•Ddenote the event that both the lightbulbs we selected are defective.•Bdenote the event that the lightbulb we selected is from Boxi.i0We wish to computeP[B|D]. Using Bayes' Rule we get:10P[D|B]·P[B]110P[B|D]=100P[B]·P[D|B]+P[B]·P[D|B]112210099··0.51000999=10099100990.5··+0.5··100099920001999=0.8
2Solve the RainbowYour roommate was having Skittles for lunch and they offer you some. There are five differentcolors in a bag of Skittles: red, orange, yellow, green, and purple, and there are 20 of each color.You know your roommate is a huge fan of the green Skittles. With probability 1/2 they ate all ofthe green ones, with probability 1/4 they ate half of them, and with probability 1/4 they only ate [Show Less]