概率数学案例 Task1 involves the Binomial&Normal distributions; task 2 involves conditional probability& the third is on integration.

Introduction
This coursework is made up of three tasks: the first involves the Binomial and Normal distributions; the second task involves conditional probability and the third is on integration. Make sure that all that working is shown and that your explanations are well presented. Marks will be given for answers and explanations that are clear and concise.

You will need to write a report on what you have discovered. Make sure that your report is informative and brief. The person who assesses your coursework will probably not be impressed by a report that runs to several hundred words when everything could have been presented in a few sentences – and you are unlikely to gain any extra marks!

Marks for each part are shown in square brackets.

Finally, when carrying out this task, take great care to check that figures are entered accurately into your calculator or computer. Just one mistake could make a big difference to your results and cause you to lose many marks.


Task 1

A merchant sells large numbers of hens’ eggs at a market. These eggs come in different sizes and a ‘medium’ sized egg will usually have a mass of between 50 and 60 grams. Unfortunately, there is little consistency in egg size: some consignments may have a large proportion of ‘medium’  sized eggs while others may have a much smaller proportion.



The merchant sells the eggs in boxes and a box can hold as many as 120 eggs. It is important that she has some idea of what proportion of the eggs in a box are ‘medium’ sized.

The size of egg has two outcomes (‘medium’ or not ‘medium’)and will follow a Binomial distribution with probability p. For a box of n eggs, the probability of x or less ‘medium’ eggs can be found using the Binomial cumulative distribution table.



From the table n = 15, choose a value of p which should be greaterthan
0.2 and any 2 different values of x from the left-hand column. Work out p(X < x) for each of your chosen values of x (If the answer is 1.000, or very close to it, then choose another value for x).[ 2 ]

Repeat part a) from the tables n = 20 and n = 25. You can use the same values of p and x if you wish but, again, you may need to change the value of x if the probability is too close to000[ 4 ]
If a box holds more than 30 eggs, using the Binomial cumulative distribution table is clearly not going to be possible. The Normal distribution can be approximated to the Binomial distribution using the mean np and the variance np(1-p), but before doing this, we have to make a continuity correction. This is because a Binomial distribution is discrete and a Normal distribution is continuous.



The continuity correction involves adding or subtracting 0.5 to the value of x e.g. for x = 6 in the Binomial distribution, we need to take x = 6.5 when using the Normal distribution to find p(X < =6) and x = 5.5 when finding p(X > =6).



c）Work out the equivalent probabilities using the Normal distribution with the continuity correction in parts [a] and [b]. What do you notice?[ 9 ]

d)Now go to the n = 30 in your Binomial cumulative distribution table and choose any four different values of p. Two of the values of p must be less than 0.5 and the other two greater than 0.5. This time, choose only one value of x and find p(X <= x) for your chosen values of p.



e)Work out the equivalent probabilities in part d) using the Normal

distribution with the continuity correction. What do you notice?



f) In view of your results, comment on the accuracy of the Normal distributionwhen usedin place of the Binomial 【2】



TASK 2

In a large college two newsletters, A and B, are issued to all the teaching staff each week. Each member of staff reads either newsletter A or newsletter B (but not both). The percentage of staff who read newsletter A is x%.



The college advertises a management post in both newsletters. It is reckoned that the probability of a reader of newsletter A applying for the post is 0.02 and the probability of a reader of newsletter B applying is 0.03.

Choose any value for x.

a）If the college receives one application, find the probability that the person who applies reads [i] newsletter A; [ii]

b）If the college receives two applications, find the probability that one applicant reads newsletter A and the other reads newsletter B.



TASK 3

It is sometimes possible to evaluate an integral by two or more different methods. One case is the integral below:

概率数学案例概率数学案例
Choose any positive value for a and any value for each of the limits h and k ,

provided k > h and h > a .

Evaluate the integral using any two suitable methods. You must show each stageof your working and answers must be given as a single expression and in exact form in both cases. [ 13 ]
Compareyour


Summary and Conclusion

Write a short conclusion (maximum of 100 words) explaining what you have discovered in these tasks. [ 6 ]