AaBbCcD AaBbl AaBbo 1 Body Text T Heading 1 1 Headin Paragraplh Styles TASK 2 Yo

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AaBbCcD AaBbl AaBbo 1 Body Text T Heading 1 1 Headin Paragraplh Styles TASK 2 You have already met the Binormial distribution and the Normal distribution. There is another distribution called the Geometric distribution which is one where you keep trying something until you succeed. An example is trying to pass a test with an unlimited number of attempts. If the probability that you pass the test is, say 0.4, then the probability that you pass the test at your third attempt is 0.6 x 0.6 x 0.4 or 0.6 x 0.4 - 0.144 (Notice that you fail the test at the first two attempts and that you need to take account of the failures up until the time you succeed.) The probability that you pass the test at the fifth attempt is 0.6*x 0.4-o.05184, at the eighth attempt is 0.6 x 0.4- 0.0112, etc. a) Take a different probability of passing the test and work out how manyI attempts are needed for the probability of first passing to be below Now take a very small probability (below 0.15) of passing the test and work out how many attemots are needed for the probability of first passing to be beloe1 b) L51 50 e a ra c) Your answers in parts [al] and [b] are based on an important assumption.[21 What is this assumption? Do you think this assumption is likely to bold o practice?

Explanation / Answer

(a)

Let's assume that probability of passing in a test is 0.25

So the probability of failing is 0.75

Let's assume you clear the test on the 'nth' attempt.

The probability of passing the test on 'nth' attemp is: (0.75^(n-1))*0.25

We want to find 'n' such that:

(0.75^(n-1))*0.25 < (1/500)

Solve to get:

n > 17.78

So, in this case you need 18 attempts.

(b)

Let's assume that probability of passing in a test is 0.11

So the probability of failing is 0.89

Let's assume you clear the test on the 'nth' attempt.

The probability of passing the test on 'nth' attemp is: (0.89^(n-1))*0.11

We want to find 'n' such that:

(0.89^(n-1))*0.11 < (1/50)

Solve to get:

n > 15.63

So, in this case you need 16 attempts.

(c)

The assumption made is that probability of passing the test is independent for each attempt.

This is not generally valid in practical sense, because once you fail a test many times, you tend to learn more and the probability of passing on the next attempt naturally increases.

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