Transcript Slide 1
CHOOSE YOUR DISTRIBUTION Binomial/Poisson/Normal 1. A typist makes on average 2 mistakes per page. What is the probability of a particular page having no errors on it? 2. The temperature my Kettle ‘boils’ water to have a mean of 960C with standard deviation of 50c. What is the chance of it ‘boiling’ water to less than 900C 3. A computer crashes once every 2 days on average. What is the probability of there being 2 crashes in one week? 4. Components are packed in boxes of 20. The probability of a component being defective is 0.1. What is the probability of a box containing 2 defective components? answers 1 A typist makes on average 2 mistakes per page. What is the probability of a particular page having no errors on it? We have an average rate here: lambda = 2 errors per page. We don't have an exact probability (e.g. something like "there is a probability of 1/2 that a page contains errors"). Hence, Poisson distribution. (lambda t) = 2. Hence P(0) = 20 e(-2) /0! = 0.135. or P.p.d x = 0 mean=2 3 A computer crashes once every 2 days on average. What is the probability of there being 2 crashes in one week? Again, average rate given: lambda = 0.5 crashes/day. Hence, Poisson. (lambda t) = (0.5 per day x 7 days) = 3.5/week & n = 2. P(2) = 3.52 e(-3.5) /2! = 0.185. or P.p.d x = 2 mean=3.5 answers 2 The temperature my Kettle ‘boils’ water to, has a mean of 960C with standard deviation of 50c. What is the chance of it ‘boiling’ water to less than 900C. z= (90-96)/5 =-1.2 (look up on table) gives 0.3849, subtract from 0.5 P(x<90) = 0.1151 Or N.c.d: lower = -99999, upper = 90, σ= 5, μ=96 4 Components are packed in boxes of 20. The probability of a component being defective is 0.1. What is the probability of a box containing 2 defective components? Here we are given a definite probability, in this case, of defective components, p = 0.1 and hence q = 0.9 = Prob. not defective. Hence, Binomial, with n = 20. faulty So P(2) = 20C2q18 p2 = 0.285 or B.p.d: n= 20 p=0.1 x=2 Binomial/Poisson/Normal 5. ICs are packaged in boxes of 10. The probability of an ic being faulty is 2%. What is the probability of a box containing 2 faulty ics? 6. The mean number of faults in a new house is 8. What is the probability of buying a new house with exactly 1 fault? 7. A box contains a large number of washers; there are twice as many steel washers as brass ones. Four washers are selected at random from the box. What is the probability that 0, 1, 2, 3, 4 are brass? 8. Washers are produce with weights that are distributed with parameters mean = 4g and standard deviation = .5g. What is the probability of a washer weighing more than 3g? answers 5 The probability of an ic being faulty is 2%. What is the probability of a box containing 2 faulty ics? We have a probability of something being true or the same thing not being true; in this case, an ic being faulty. Hence, Binomial distribution. p = P(faulty) = 0.02, q = P(not faulty) = 0.98. n = 10. x=2 So, Prob of a box containing 2 faulty ics P(2) = 10C2 q8p2 = 0.015. Or B.p.d n=10, p=0.02, x =2 6 The mean number of faults in a new house is 8. What is the probability of buying a new house with exactly 1 fault? Here we have an average rate of faults occurring: 8 per house. Hence, Poisson, with (lambda t) = (8 faults/house * 1 house) = 8. n = 1 too, so P1 = 81e (-8) /1!= 0.0027. Or P.p.d mean = 8, x = 1 7 A box contains a large number of washers; there are twice as many steel washers as brass ones. Four washers are selected at random from the box. What is the probability that 0, 1, 2, 3, 4 are brass? Here too we have a probability of brass (1/3) and of not brass --- i.e. steel --which is 2/3. Hence, use the Binomial distribution with p = 1/3 , q = 2/3 and n = 4. Use formulae for each x value No. of brass so P(0) = 0.197, P(1) = 0.395, P(2) = 0.296, P(3) = 0.099 and P(4) = 0.012. or B.c.d n=4, p = 1/3, x=0, 1, 2, 3, 4 8. Washers are produce with weights that are distributed with parameters mean = 4g and standard deviation = .5g. What is the probability of a washer weighing more than 3g? z=3-4/0.5=-2 gives 0.4772 (add 0.5) P(x>3) =0.9772 Or N.c.d lower = 3, upper = 999999, σ= 0.5, μ=4 Typical Questions Typical questions 1 A shop sells biscuits that are said to contain on average 5 chocolate chips each. a What is the probability that a biscuits contains 4 chocolate chips? b What percentage of biscuits would you expect to contain less than 3 chocolate chips? c Find the probability that half a biscuits contains exactly 2 chocolate chips. d Another type of biscuit is found to have no chocolate chips 5% of the time. What is the average number of chips per biscuit? A factory produces tyres. 5% of the tyres produced are rejected because of defects. Find the probability that in a batch of 10 tyres: a none are defective; b less than 2 are defective. c at least 3 are defective d Another company has studied the amount of defective tyre per 10 tyre batch. They found that 88% of the time they have a batch with no defects. What is the chance that an individual tyre has a defect. A multi-choice test has 10 questions with 4 options for each. If I guess every answers, find the chance of A) getting them all wrong B) getting at least 1 correct C) passing (50% or more) D) A teacher wants to set a test similar to the above one that has a less than 0.5% chance of a student guessing and getting all questions correct. What is the least number of questions needed A test has a mean of 60% with a standard deviation of 12%. Find the chance of: A) scoring less than 30% B) passing the test What would you need to score above to C) Put yourself in the top 20% I get sick 2.8 times per year, on average. Find the probability of: A) me not getting sick this year B) getting sick this month Another person finds they get sick at least once in a year 65% of the time C) How many times do they get sick per year.