M2S1: Probability and Statistics II
Problem Sheet 1
Questions marked (†) are revision of M1S
M2S1概率与统计代写 1.(†)Find formulas for each of the following events in terms of α = Pr(A), β = Pr(B), and γ = Pr(A ∩ B). a）either A or B or both. b）···
1.(†)
Find formulas for each of the following events in terms of α = Pr(A), β = Pr(B), and
γ = Pr(A ∩ B).
a）either A or B or both.
b）either A or B but not both.
c）at least one of A orB.
d）at most one of A orB.
2.(†) M2S1概率与统计代写
A couple plans to have three children. There are 8 possible arrangements of girls and boys. For example, GGB means the first two children are girls and the third child is a All 8 arrangements are (approximately) equally likely.
a）Write down all 8 arrangements of the sexes of the three children. What is the probabilityof any one of these arrangements?
b）LetX be the number of girls the couple What is the probability that X = 2?
c）Starting from your work in (a), find the distribution of X. That is, what values can X take, and what are the probabilities for each value?
d）What are the mean and variance of X?
3.(†) M2S1概率与统计代写
If Pr(A) = 1/4 and Pr(Bc) = 1/5, can A and B be disjoint? Explain.
4.(†)
Suppose X is a set containing n If A and B are randomly chosen subsets of X, what is the probability that A ⊆ B?
5.(†)
Consider the probability space (S , F, Pr) with A, B. Using only the Kolmogorov axioms prove
a）Pr(A) ≤1,
b）IfA ⊂ B, then Pr(A) ≤ Pr(B), and
c）Pr(A∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B).
6.(†) M2S1概率与统计代写
Let S be a set.
a）Showthat the collection F = {∅, S} is a sigma algebra.
b）Showthat for any subset E ⊆ S, FE = {∅, E, Ec, S} is a sigma algebra.
c）Let F be the collection of all subsets of S. Show that F is a sigma algebra.
d）Show that the intersection of two sigma algebras on S is a sigma algebra.
e）Givean example to show that the union of two sigma algebras on S need not be a sigma algebra.
7.(†) M2S1概率与统计代写
Let S = [0, 1], the unit interval. Define F to be the collection of all countable or co-countable subsets of S , where a co-countable set is one whose complement is countable. Show that F is a sigma algebra. [Hint: Is a countable union of countable sets countable?]
For discussion
Choose one or two questions to discuss in groups
8.(†)
Acity contains two hospitals, one small and one The long-term proportion of boys born can be taken to be 50%. On a given day, the proportion of boys born in one of the hospitals is 55%. Which hospital is this more likely to be?
9.(†) M2S1概率与统计代写
Tom is an opera buff who enjoys touring art museums when on holiday. Growing up, heenjoyed playing chess with family members and friends. Which of the following two situations is more likely?
a）Tom plays trumpet for a major symphony orchestra.
b）Tom is a farmer.
Data question
10.(†) M2S1概率与统计代写
Consider the data in Table 1, comparing two treatments for kidney stones.
Table 1: Success rates for the treatment of kidney stones by open surgery (Treatment A) and percuta- neous nephrolithotomy (Treatment B).
| Treatment A | Treatment B | |
| Small stones
Large stones |
81 / 87 (93%)
192 / 263 (73%) |
234 / 270 (87%)
55 / 80 (69%) |
| Total | 273 / 350 (78%) | 289 / 350 (83%) |
Which treatment is better? Evaluate the following two responses:
I. 83% of the time, Treatment B was successful, whereas Treatment A was successful only 78% of the time. So Treatment B is better.
II. For patients with small stones, Treatment A was 93% successful whereas Treatment B was 87% successful. Similarly, for large stones, Treatment A was 73% successful, while treatmentB was 69% successful. So Treatment A is better.
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