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UCLA Statistics, Neurology, LONI, Math/PIC

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STAT 35 Spring 2008	*Interactive and Computational Probability*
Department of Statistics	Instructor: Ivo Dinov
Homework 3	Due Date: Friday, May 09, 2008

Please, submit your homework right after lecture on the due date. See the HW submission rules. On the front page include the following header.

(HW_3_1) Shortly after being put into service, some buses manufactured by a certain company have developed cracks on the underside of the main frame. Suppose a particular city has 20 of these buses, and cracks have actually appeared in 8 of them.

How many ways are there to select a sample of 5 buses from the 20 for a thorough inspection?
In how many ways can a sample of 5 buses contain exactly 4 with visible cracks?
If a sample of 5 buses is chosen at random, what is the probability that exactly 4 of the 5 will have visible cracks?
If buses are selected as in part (c), what is the probability that at least 4 of those selected will have visible cracks?

(HW_3_2) Three molecules of type A, 3 of type B, 3 of type C, and 3 of type D are to be linked together to form a chain molecule. One such chain molecule is ABCDABCDABCD, and another is BCDDAAABDBCC.

How many such chain molecules are there?
Suppose a chain molecule of the type desired is randomly selected. What is the probability that all three molecules of each type end up next to one another (such as BBBAAADDDCCC)?

(HW_3_3) A certain sports car comes equipped with either an automatic or a manual transmission, and the car is available in one of four colors. Relevant probabilities for various combinations of transmission type and color are given in the table below. Let A = {automatic transmission} , B = { black } , and C = { white }.

		Color
		White	Blue	Black	Red
Transmission Type	A	0.15	0.10	0.10	0.10
Transmission Type	M	0.15	0.05	0.15	0.20

(a) Calculate P(A), P(B), and P(A ∩ B).
(b) Calculate both P(A | B) and P(B | A), and explain in context what each of these probabilities represent.
(c) Calculate and interpret P(A | C) and P(A | C').

(HW_3_4) Use the Die Coin Experiments, from SOCR Experiments, to emperically validate the Baysian rule for computing conditional probabilities:

Suppose D3 = event { Die turns up a three (first part of the experiment) } and 2H = event { There are two Heads observed in the tossed coins (second part of the experiment) }. Run the experiment 100 times first with flat-die-probabilities (p=1/6) and a fair-coin (p=0.5). By counting outcomes of interest validate computationally that:

P( D3 | 2H ) = [ P( 2H | D3 ) P( D3 ) ] / P( 2H )

Now run the experiment 100 times first with a loaded die with probabilities (p1= 0.01, p2=0.05, p3=0.1, p4=0.2, p5=0.34, p6=0.3) and a loaded coin (p=0.3). Again by counting outcomes of interest validate computationally that:

P( D3 | 2H ) = [ P( 2H | D3 ) P( D3 ) ] / P( 2H )

(HW_3_5) Show emperically the following approximations: Normal to Binomial, Binomial to Hypergeometric, Normal to Poisson and Poisson to Binomial. Use SOCR Distributions and SOCR Experiments. Can you find emperical situations when these approximations fail? Please provide specifics, snapshots, try to explain why, etc.

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