Question

1. Consider a biased dice, where the probability of rolling a 3 is 4 9 . The dice is rolled 7 times. If X denotes the number of 3’s thrown, then find the binomial distribution for x = 0, 1, . . . 7 and complete the following table (reproducing it in your written solutions). Give your answers to three decimal places.

x	0	1	2	3	4	5	6	7
Pr(X=x)

2. The Maths Students Society (AUMS) decides to conduct small lottery each week to raise money. A participant must pays $2 to enter and chooses three distinct numbers between 1 and 10 (the order does not matter). If their three chosen numbers match the three numbers drawn by AUMS they win a $70 jackpot offered each week, otherwise they recieve nothing. No two entries can use the same numbers.

(a) How many distinct entries can there be?

(b) Write out the probability distribution of returns for one random entry?

(c) A student enters 15 times in one week with different sets of numbers. Determine their probability of winning and the expected return.

(d) A different student enters once each week for 12 weeks. Determine their probability of winning at least once and the expected return.

(e) Suppose 20 entries are made every week for a year. By first calculating the expected amount raised each week, determine how much money AUMS expects to raise in a year?

31. Consider the function f(x) = 3x⁴ − 8x³ + 1.

(a) Find the derivative f 0 (x), and hence the critical points for the function (for this question give both x and y coordinates).

(b) Classify the critical points using the first derivative test to determine if they are local maximum or minimum or neither.

(c) Find any points of inflection for f(x) (give both x and y coordinates).

Answer 1

1. The formula to be used for this question is given by the pdf of the binomial distribution which is given by the formula

where n- no. of trials

p-probability of success, here, getting a three

and q-probability of failure, here, not getting a three.

2)

a) Here, no two entries can use the same number. In total we have to choose 3 numbers from 10 numbers such that no two individuals can use the same set of numbers.And hence, the number of distinct entries is given by 10C3=120

b)Let X denote the random variable of the returns. X can either have the value 0 or 70. The probability that X=70,

P(X=70)=1/120

P(X=0)=119/120

Because it has to match with all th three numbers selected at random by AUMS.

c) Here the student enters 15 times in a week. So, the student has a chance to win $70 if any of these 15 triplets matches with that selected by AUMS. And this probability is P(1st one matches or second one matches+...15th one matches)and it is given by 1/120+1/120+...1/120(15 times)=15/120

Expected return=

=70*15/120+0*105/120=$8.75

d) Here a student enters the contest for 12 weeks. Each week can be considered as a bernoulli trial with p=1/120 and q=119/120

The probability when n=12 is 12Cx(1/120)^x *(119/120)^n-x

The probability of winning atleast once is same as 1-probability of not even one success.

i.e. 1-P(x=0)=1-0.9044=0.09554

e)There are52 weeks in a year. If 20 entries are made per week, the chance of the organisers losing $70 becomes 20/120.

And the total money raised per week is $40.It will be $2080 per year. Let x denote thenumber of times a match is made. X can have 1-52 values.

The expected amount is given by

=

3.

Here,   ------------(A)

------------(1)

Equating to zero and solving eqn (1) , we get the critical points as x=0 and x=2.

Therefore , substituting these values in eqn (A), we get y=1 and y=15.

So the critical points are (0,1) and (2,15).

b) Now we need to classify them as local maximum or local minimum or neither.

For this we need to fin the second derivative,

---------(2)

Substituting x=0 in (2) we get =0. Hence it is neither a point of local maximum or local minimum.

And . Hence (2,15) is a point of local minimum.

c) The point of inflection is (0,1)