Question

In: Statistics and Probability

Dr. Palpatine teaches statistics at Coruscant University. He believes there are three types of classes he...

Dr. Palpatine teaches statistics at Coruscant University. He believes there are three types of classes he may encounter:

(P) Poor classes, where only 70% of the students will be able to pass exam 1 in the course.

(A) Average classes, where 85% of the students will be able to pass exam 1.

(G) Good classes, where 95% of the students will be able to pass exam 1.

Assuming Palpatine teaches a class of 35 students...

If the class is poor (P), what is the probability that exactly 30 of them will pass (E), that is what is P(E|P)? _______

If the class is average (A), what is the probability that exactly 30 of them will pass (E), that is what is P(E|A)? _______

If the class is average (G), what is the probability that exactly 30 of them will pass (E), that is what is P(E|G)? _______

Now suppose that Palpatine initially believed the probability that his class was poor, average, and good was 25%, 50%, and 25% respectively. Use Bayes' Rule and the results above to find P(G|E), the probability his class is a good one (G) given the evidence E that exactly 30 of them passed the first exam. ______

Expert Solution

1. P (E | P)

We can compute above probability by considering given situation as a Binomial Experiment with "Success" defined as "A student passes the exam"; and success probabilities for POOR class p = 0.70, Total trials N = 35.

So, P( E | P ) = P( 30 success in Binomial experiment with N=35, p = 0.70) = = 0.01778

2. P(E | A) = P( 30 success in Binomial experiment with N=35, p = 0.85) = = 0.1881

3. P ( E | G ) = P( 30 success in Binomial experiment with N=35, p = 0.95) = = 0.02177

4. Now, from Bayes Rule : P ( G | E ) = (P( E | G ) * P ( G ) ) / P( E )

P ( G ) = 0.25; P ( E | G ) = 0.02177;

From total probability theorem, P ( E ) = P(E | G ) * P(G) + P( E |A ) * P ( A ) + P ( E | P) * P ( P )

= 0.02177*0.25 + 0.1881*0.50 + 0.01778*0.25

P ( E ) = 0.10393

Therefore, P ( G | E ) = (0.02177 * 0.25) / 0.10393 = 0.052366

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orchestra answered 1 year ago

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