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# English

Submitted By Johnnywang
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Pages 3
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North Sydney Campus

Semester 2, 2014

Probability Trees

A survey of STAT102 students reveals that among the students that attain CR or better grades, 60% of them attended both lectures and tutorials. Among the students that attain PASS or FAILURE, only 20% of them attended both lectures and tutorials.

Not attending both lectures and tutorials and with self-study, a student feels that the chance of attaining CR or better grades is quite low, 10%. The student is willing to attend both lectures and tutorials only if the efforts would increase the probability of achieving CR or better grades to 20% or more.

Please use the following notations for the various states:

C = CR or better grades

CC = PASS or FAILURE

L = Attend both lectures and tutorials

LC = Do not attend both lectures and tutorials

Required

Should the student attend both lectures and tutorials? Please justify your answer with the posterior probability (prior probabilities are revised after the decision to attend both lectures and tutorials).

The prior probability (probabilities determined prior to the decision of attending both lectures and tutorials) of attaining CR or better grades is

P(C) = 0.1

P(CC) = 0.9 Complement rule

Conditional probabilities are

P(L(C) = 0.6 (for a student that have achieved a CR or better grade, the probability of attending both lectures and tutorials is 0.6)

P(L(CC) = 0.2

Applying complement rule, the following conditional probabilities are

P(LC(C) = 1 - 0.6 = 0.4 (for a student that have achieved a CR or better grade, the probability of not attending both lectures and tutorials is 0.6)

P(LC(CC) = 1 - 0.2 = 0.8

|Probability Tree | | |
|Prior Probability |Likelihood Probability|Joint Events |Joint Probabilities | | |
| | | | | | |
| |L(C 0.6 |L and C |0.1 x 0.6 = 0.06 | | |
|C 0.1 | | | | | |
| |LC(C 0.4 |LC and C |0.1 x 0.4 = 0.04 | | |
| | | | | | |
| |L(CC 0.2 |L and CC |0.9 x 0.2 = 0.18 | | |
|CC 0.9 | | | | | |
| |LC(CC 0.8 |LC and CC |0.9 x 0.8 = 0.72 | | |
| | | | | | |

We need to compute the probability that the student would attain CR or better grades given that the student attended both lectures and tutorials, viz.

P(C(L)

Using the definition of conditional probability:

P(C(L) = P(C and L) / P(L)

P(C and L) = 0.1 x 0.6 = 0.06

P(CC and L) = 0.9 x 0.2 = 0.18

P(L) = P(C and L) + P(CC and L) = 0.06 + 0.18 = 0.24

Thus

P(C(L) = 0.06 / 0.24 = 0.25

If the student attends both lectures and tutorials, the probability of attaining CR or better grades is 0.25, which is greater than 0.2, the threshold probability for attaining CR or better grades if attending both lectures and tutorials. Therefore, the student should attend both lectures and tutorials.

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