1. Assume that the sample space is Ω = fall non-negative integers that are less or equal to 10g, i.e. Ω = f0; 1; 2; : : : ; 10g; and each outcome is equally probable,i.e. P(w) = 11 1 for any w 2 Ω. We also define two events:

A = fall prime numbers within Ω.g

B = fall even numbers within Ω.g

Note: a prime number is an integer that is greater than 1 and has no positive integer divisor other than 1 and itself. For example, 11 is a prime as it can only be divided by 1 and 11 (itself); but 12 is not as 12 = 3 ∗ 4 so other than 1 and 12, it is also divisible by 3 and 4.

a) Write down all the possible outcomes of the events A and B.

b) Calculate the values of P(A \ B) and P(A [ B).

c) Calculate the value of P(A｜B).

c) Calculate the value of P(B｜A¯).

2. Suppose that 10% of undergraduate students at a university are foreign students,and that 25% of the postgraduate students are foreign students. Also, assume there are four times as many undergraduates as postgraduate students. Let Ω = fall students in the universityg and define the following events,

U = fundergraduate students of the universityg
P = fpostgraduate students of the universityg

F = fforeign students of the universityg
D = fdomestic students of the universityg

Obviously, U and P form a partition of Ω, and F and D form a partition of Ω:

a) Write down all four probability statements based on the information pro vided. (Note: Your first statement should be P(F jU) = ? with ? evaluated.)

b) What proportion of the foreign students are undergraduate? Show your working.

c) Are F and U statistically independent? Why or why not?

3. It is known that 60% of a certain species of NZ native birds have a distinctive characteristic A. The Department of Conservation (DoC) ran an experiment on a new sanctuary zone. Twelve birds of this species are captured randomly and independently in that zone; and three of them are identified to have character istic A.

One of the DoC researchers want to determine if it is reasonable to assume the birds in this new sanctuary zone have a different probability than the species has in general. He runs a two-sided hypothesis test. Let X be the number of birds with characteristic A from a random sample of 12 native birds of that species.

a) Write down the null hypothesis and the alternative hypothesis.

b) Specify the distribution of X under the null hypothesis H0. Remember to specify the parameter value(s) of the distribution.

c) Calculate the expected value and variance of X, i.e. E(X) and Var(X).

d) Calculate the probability of capturing exactly 3 birds with characteristic A out of a sample of 12 birds.

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