Question

# Suppose that a university student has to take 10 independent courses in the final year. The class of honors awarded is based on the results of the final year examination. If a student gets A‐ or above for at least 8 courses, he/she will be awarded a first

Suppose that a university student has to take 10 independent courses in the final year. The class of honors awarded is based on the results of the final year examination. If a student gets A‐ or above for at least 8 courses, he/she will be awarded a first class honors.

Under ordinary circumstances, John will get A‐ or above for at least 8 courses with probability of 0.2. However, with a probability of 0.3, John will be in a low mood during the examination period. Under that low mood case, John will get A‐ or above for each course with probability 0.45.

(a) What is the probability that John will get A‐ or above for at least 8 courses given that he is in low mood during the examination period?

(b) What is the probability that John will achieve a first class honors?

a)

A distribution is a binomial distribution when

1) All trials are identical and independent.

2) two outcomes for each trial: Success, failure

3) Number of trials is fixed.

Here we have 10 courses. Each course is identical and independent. For each course, we have two outcomes: A- or above, below A-.

For low mood condition probability, John will get A‐ or above for each course is 0.45.

So This distribution is binomial with n=10 and p=0.45

Here at least 8 A- or above means we get A- or above in 8 or 9 or 10 courses.

So P(at least 8 A-)= P(8 of 10)+P(9 of 10)+P(10 of 10)

for a binomial distribution with n independent trials with probability p, equation to find the probability of k success in n trials is

Using this idea

P(at least 8 A-)=0.02288959+0.00416174+0.0003405=0.02739183

(round according to question)

P(at least 8 A- | low mood)=0.02739183

b)

For two events A and B we have P(A and B)=P(A)* P(B)

Here we need to find P(John achieves first-class).

This will be honored if and only if he gets at least 8 A-.

We have two conditions here: ordinary and low mood.

P(low mood)=0.3

Hence P(ordinary)= 1-0.3=0.7

In both situations, we have a chance for getting a first-class. So the sum of getting At least 8 A- in ordinary and low mood situations gives the probability of getting honored by first class. So

P(John achieves first-class)= P(ordinary and at least 8A-)+P(low mood and at least 8A-)

Ordinary

Here probability of at least 8 A- is 0.2. So P(at least 8 A- | ordinary)= 0.2

Then P(ordinary and at least 8A-)= P(at least 8 A- | ordinary) *P(ordinary)= 0.2*0.7=0.14

Low mood

We founded P(at least 8 A- | low mood)=0.02739183

So

P(low mood and at least 8A-)= P(at least 8 A- | low mood) *P(low mood)=0.02739183*0.3=0.008217549

So

P(first class)= 0.14+0.008217549= 0.148217549

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