Math Answers

If there are 21 guests and she divides both types of truffles equally among them, then each guest will get chocolate truffles and caramel truffles.

3Fn − Fn−2 = Fn+2, for n ≥ 3.

Let r=xi^+yj^+zk^ and r=||r||. Show that:

∇(lnr)=r/r^2.

and

∇×((r^n)r)=0.

Before an increase in excise duty on tea, 800 people out of a sample of 1000 were

consumers of tea. After the increase in duty, 800 people were consumers of tea in a

sample of 1200 persons. Write down the R programming code to test whether the

significant decrease in the consumption of tea after the increase in duty at 1 % level of

significance.

If A(u) is a differentiable vector function of u and ||A(u)||=1 , prove that dA/du is perpendicular to A .

The dean of a university wants to use the mean of a random sample to estimate the average amount of time students take to get from one class to the next. She wants to assert with probability 0.95 that her error will be at most 0.25 minutes. If she knows from the past studies that population standard deviation is 15 minutes, how large a sample she will need?

Amir and Jingpeng each play a single round of a computer game, which ends with a win or a loss. Let A denote the event that Amir loses, and J denote the event that Jingpeng loses the game they play, respectively. Suppose that P(A) = 1/9 and P(J) = 1/12. We assume that the events A and J are independent.

Find the probability that:

(i) Jingpeng wins the game it plays.

(ii) Both of them lose the game they play.

(iii) At least one of them loses the game they play.

(iv) Both of them win the game they play.

(i) How many ways can you save the files into the 6 folders if the temporal order in which

you save the individual files does not matter.

(ii) How many ways can you save the files into the 6 folders if the temporal order in which

you save the individual files matters.

State how you would interpret the relationship between two variables whose sample correlation coefficient r is calculated to be

1. r = 0.985

2. r = - 0.76

3. r = 0.001

You have a deck of 52 playing cards.

(i) How many different 8 card hands can be dealt?

(ii) What is the probability that a hand of 8 dealt randomly contains (exactly) 2 aces?

(iii) What is the probability that a hand of 7 dealt randomly will have 7 cards of the same

suit?

An influenza vaccine is produced by two different companies. It is known that a vaccine

produced by company 1 is effective with probability 0:89, while a vaccine produced by

company 2 is effective with probability 0:93. We also know that company 1 supplies 40%

of the vaccines, while company 2 supplies 60% of the vaccines ordered by the government.

(i) What is the probability that a vaccine is effective, given that it was produced by company2?

(ii) What is the probability that a randomly chosen vaccine from the government’s order

is not effective?

(iii) What is the probability that given a vaccine is not effective that it was produced by

company 1?