1. CHOOSE YOUR DISTRIBUTION

2. 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 96 0 C with standard deviation of 5 0 c. What is the chance of it ‘boiling’ water to less than 90 0 C  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?

3. 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) = 2 0 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) = 3.5 2 e (-3.5) /2! = 0.185. or P.p.d x = 2 mean=3.5

4. answers  2 The temperature my Kettle ‘boils’ water to, has a mean of 96 0 C with standard deviation of 5 0 c. What is the chance of it ‘boiling’ water to less than 90 0 C. 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) = 20 C 2 q 18 p 2 = 0.285or B.p.d: n= 20 p=0.1 x=2

5. 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?

6. 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) = 10 C 2 q 8 p 2 = 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 = 8 1 e (-8) /1!= 0.0027. Or P.p.d mean = 8, x = 1

7.  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

8. Typical Questions

9. Typical questions 1A shop sells biscuits that are said to contain on average 5 chocolate chips each. aWhat 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. dAnother type of biscuit is found to have no chocolate chips 5% of the time. What is the average number of chips per biscuit?

10. A factory produces tyres. 5% of the tyres produced are rejected because of defects. Find the probability that in a batch of 10 tyres: anone are defective; bless than 2 are defective. cat least 3 are defective dAnother 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.

11.  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

12.  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%

13.  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.