Probability Conditional Exercises

Solutions in {curly brackets}. Worked solution clips can be found here Q1, Q2, Q3, and Q5.

1. [math]A[/math] and [math]B[/math] are events such that [math]\Pr (A)=0.4[/math] and [math]\Pr (A\cup B)=0.75.[/math]

   1. Find [math]\Pr (B)[/math] if [math]A[/math] and [math]B[/math] are mutually exclusive. {0.35}

   2. Find [math]\Pr (B)[/math] if [math]A[/math] and [math]B[/math] are independent. {0.58333}

2. Events [math]A,[/math] [math]B[/math] and [math]C[/math] are such that [math]B[/math] and [math]C[/math] are mutually exclusive and [math]\Pr (A)=2/3,[/math] [math]\Pr (A\cup B)=5/6[/math] and [math]\Pr (B\cup C)=4/5[/math]. You also know that [math]\Pr (B|A)=1/2[/math] and [math]\Pr (C|A)=3/10[/math]. Is there reason to believe that [math]A[/math] and [math]C[/math] are not statistically independent? {no as [math]\Pr(C)=\Pr (C|A)[/math]}

3. A sample of 1000 undergraduates were asked whether they took either Mathematics, Physics or Chemistry at A-level. The following responses were obtained: 100 just took Mathematics; 70 just took Physics; 100 just took Chemistry; 150 took Mathematics and Physics, but not Chemistry; 40 took Mathematics and Chemistry, but not Physics; and, 240 took Physics and Chemistry, but not Mathematics. Calculate the following:

   1. Of those who took Mathematics, what proportion also took Physics (but not Chemistry) and what proportion took both Physics and Chemistry? {(150/(290+x)) and (x/(290+x)) where [math]0 \leq x \leq 300[/math]}

   2. Of those who took Physics and Chemistry, what proportion also took Mathematics? {(x/(240+x)) where [math]0 \leq x \leq 300[/math]}

4. The Survey of British Births, undertaken in the 1970s, aimed to improve the survival rate and care of British babies at, and soon after, birth by collecting and analysing data on new-born babies. A sample was taken designed to be representative of the whole population of British births and consisted of all babies born alive (or dead) after the 24th week of gestation, between 0001 hours on Sunday 5 April and 2400 hours on Saturday 11 April 1970. The total number in the sample so obtained was [math]n=17,530.[/math] A large amount of information was obtained, but one particular area of interest was the effect of the smoking habits of the mothers on newly born babies. In particular, the ability of a newly born baby to survive is closely associated with its birth-weight and a birth-weight of less than 1500g is considered dangerously low. Some of the relevant data are summarised as follows.

   For all new born babies in the sample, the proportion of mothers who:(i) smoked before and during pregnancy was 0.433(ii) gave up smoking prior to pregnancy was 0.170(iii) who had never smoked was 0.397.

   However, by breaking down the sample into mothers who smoked, had given up, or who had never smoked, the following statistics were obtained:(iv) [math]1.6\%[/math] of the mothers who smoked gave birth to babies whose weight was less than 1500g,(v) [math]0.9\%[/math] of the mothers who had given up smoking prior to pregnancy gave birth to babies whose weight was less than 1500g,(vi) [math]0.8\%[/math] of mothers who had never smoked gave birth to babies whose weight was less than 1500g.

   1. Given this information, how would you estimate the risk, for a smoking mother, of giving birth to a dangerously under-weight baby? What is the corresponding risk for a mother who has never smoked? What is the overall risk of giving birth to an under-weight baby?

   2. Of the babies born under [math]1500[/math]g[math],[/math] estimate the proportion of these (a) born to mothers who smoked before and during pregnancy; (b) born to mothers who had never smoked.

   3. On the basis of the above information, how would you assess the evidence on smoking during pregnancy as a factor which could result in babies being born under weight?

5. Metal fatigue in an aeroplane’s wing can be caused by any one of three (relatively minor) defects, labelled [math]A,\,\,B\,\,[/math]and [math]C,[/math] occurring during the manufacturing process. The probabilities are estimated as: [math]\Pr (A)=0.3,[/math] [math]\Pr (B)=0.1,[/math] [math]\Pr (C)=0.6.[/math] At the quality control stage of production, a test has been developed which is used to detect the presence of a defect. Let [math]D[/math] be the event that the test detects a manufacturing defect with the following probabilities: [math]\Pr (D|A)=0.6,[/math] [math]\Pr (D|B)=0.2,[/math] [math]\Pr (D|C)=0.7.[/math] If the test detects a defect, which of [math]A,[/math] [math]B[/math] or [math]C[/math] is the most likely cause? (Hint: you need to find, and compare, [math]\Pr\left( A|D\right)[/math], [math]\Pr \left( B|D\right) [/math] and [math]\Pr \left( C|D\right) [/math] using Bayes Theorem.) {C is most likely, then A and then B. A is 9 times more likely than B, and C is 21 times more likely than B}