Hard Probability Questions

[khalil]

The Essential Mathematics book gives REALLY HARD probability questions...which leads to my question extracted from the first chapter of probability from the review (I'm having trouble answering 'c'):


A given investment scheme is such that there is a 10% chance of receiving a profit of 40%
of the amount invested, a 15% chance of a 30% profit, a 25% chance of a 20% profit, a 20%
chance of a 10% profit, a 15% chance of breaking even, a 10% chance of a 10% loss and a
5% chance of 20% loss.

a Find the mean and standard deviation of the percentage return on the amount invested.

b Find the probability that the percentage return on the amount invested is within two
standard deviations of the mean.

c An investor investing in the scheme pays a brokerage fee of 2% on the amount invested
and a tax of 40% on the return (= profit − brokerage) of the investment (assume that a
loss results in a tax refund for this investment).

Express the percentage gain in terms of the percentage return on the amount invested
and hence find the mean and standard deviation of the percentage gain.

[hyperblade01]

Basically for c, you have to find new P's


Start off with the 40% one.
Let x be the original amount invested, therefore 40% profit is 0.4x.
There is a brokerage fee of 2%, which is 0.02x
There is a tax on the return (0.4x-0.02x) * 0.4

New percentage return is:
Return - brokerage fee - (profit - brokerage)*0.4
1.4x - .02x - (0.4x-0.02x)*0.4
= 1.228x

Which means, the new return is 22.8%



Do it for every value, the only thing you change is return and profit, when you get to loss change the taxed part to positive.


OLD: 40%    30%    20%    10%    0        -10%    -20%

NEW: 22.8% 16.8%  10.8% 4.8%  -1.2%  -7.2%   -13.2%

[khalil]

Basically for c, you have to find new P's


Start off with the 40% one.
Let x be the original amount invested, therefore 40% profit is 0.4x.
There is a brokerage fee of 2%, which is 0.02x
There is a tax on the return (0.4x-0.02x) * 0.4

New percentage return is:
Return - brokerage fee - (profit - brokerage)*0.4
1.4x - .02x - (0.4x-0.02x)*0.4
= 1.228x

Which means, the new return is 22.8%



Do it for every value, the only thing you change is return and profit, when you get to loss change the taxed part to positive.


OLD: 40%    30%    20%    10%    0        -10%    -20%

NEW: 22.8% 16.8%  10.8% 4.8%  -1.2%  -7.2%   -13.2%

How did you get from 1.228x to 22.8%?

[khalil]

A game of chance consists of rolling a disc of diameter 2 cm on a horizontal square board. The board is divided into 25 small squares of side 4cm. A player wins a prize if, when a disc settles, it lies entirely within any one small square. There is a ridge round the outside edge of the board so that the disc always bounces back, cannot fall off and lies entirely within the boundary of the large square.

Prizes are awarded as follows:

centre (the middle square) 50c

inner (the eight squares surrounding the centre) 25c

corner (the four corner squares) 12c

outer (any other smaller square) 5c

When no skill is involved, the centre of the disc may be assumed to be randomly distributed over the accessible region.
a Calculate the probability in any one throw of winning:
i 50c ii 25c iii 12c iv 5c v no prize

b The proprietor wishes to make a profit in the long run, but is anxious to charge as little
as possible to attract customers. He charges C cents, where C is an integer. Find the
lowest value of C that will yield a profit.

Why do they assume the ridge has length 1 cm?

[hyperblade01]

How did you get from 1.228x to 22.8%?

You went from 1x to 1.228x, which is a profit of 22.8%

[khalil]

Monique is practising goaling for netball. She knows from past experience that her chances
of making any one shot is about 70%. Her coach has asked her to keep practising until she
scores 50 goals. How many shots would she need to attempt to ensure that the probability
of making at least 50 shots is more than 0.99?

How do I go about this?

[TrueTears]

lol I remember this good ol q, you need to use the "table method" on your TI-89 calc. Do you know how?

[khalil]

I use the CAS calc...but i could interpret your TI-89 steps to use on my cas ti-nspire

[ryley]

TrueTears, would you be able to quickly go through how to do that question, I remember having problems with it a few months ago, but I forgot about it completely.