## Thursday, December 16, 2010

### Binomials

Question 32,
Military radar and missile detection systems are designed to warn a country of an enemy attack.  A reliability question is whether a detection system will be able to identify an attack and issue a warning.  Assume that a particular detection system has .90 proabability of detecting a missile attack.  Use the binomial probability distribution to answer the following questions.
B) If two detection systems are installed in the same area and opertate independantly, what is the probability that at least one of the systems will detect the attack?
C) if three systems are installed, what is the probability
I’ve drawn a beautiful tree diagram to illustrate.

B: two systems operating independently, so it is like tossing a coin. What happens to the first system doesn’t affect the second system. Look at the branches in the trees that show whether there has been a successful detection. There are THREE (at least one). So the probability is 0.81 + 0.09 + 0.09 = 0.99. In binomials, we want a cumulative probability. There are different ways of solving this. First off, think of what is the probability of none of the systems working? That is 1-0.99= 0.01. In binomials, =1-binomdist(0,4,0.9,false).
For three systems,  extend the tree diagram with extra branches. Calculate by hand using the tree diagram and also using binomdist (It will help, I promise!).

Question 36
According to a survey conducted by TD Ameritrade, one out of four investors have exchange-traded funds in their portfolios.  For a sample of 20 investors, answer the following questions.
A) compute the probability that atleast 4 investors have exchange-traded funds in their portfolis?
B) compute the probability of atleast 2?
D) compute the expected number of investors who have exchang-traded funds in their portfolis?
A: =1 – binomdist(4,20,0.25,true) get the 0.25 from one in four investors
B: =1-binomdist(2,20,0.25,true)

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