Question

The odds that a thesis will be favorably reviewed by 3 independent examiners are 5:2, 4:3 and 3:4 respectively. What is the probability that, of the three examiners
a) all reject the thesis?
b) all approve the thesis?
c) majority approve the thesis?

Answer 1

Solution:

Since odds are in favour,let the event of examine and reviewed the thesis by three examiner's be E1,E2,E3

p(E1) = 5/7

p(E2) =4/7

p(E3)=3/7

The examiner Not passed the thesis

$p(\bar E1) = 2/7\\\\p(\bar E2) =3/7\\\\p(\bar E3)=4/7$

a) all reject the thesis?

$p(\bar E1) \times p(\bar E2) \times \: p(\bar E3) \\ \\ = \frac{2}{7} \times \frac{3}{7} \times \frac{4}{7} \\ \\ = \frac{24}{343}$

b) all approve the thesis?

$= p(E1) \times p(E2) \times p(E3) \\ \\ = \frac{5}{7} \times \frac{4}{7} \times \frac{3}{7} \\ \\ = \frac{60}{343}$

c) majority approve the thesis?

$p(E1) \times p(E2) \times p(E3) +p(\bar E1) \times p(E2) \times p(E3) + p(E1) \times p(\bar E2) \times p(E3) + p(E1) \times p(E2) \times p(\bar E3) \\ \\ = \frac{5}{7} \times \frac{4}{7} \times \frac{3}{7} + \frac{2}{7} \times \frac{4}{7} \times \frac{3}{7} + \frac{5}{7} \times \frac{4}{7} \times \frac{3}{7} + \frac{5}{7} \times \frac{4}{7} \times \frac{4}{7} \\ \\ = \frac{60}{243} + \frac{24}{343} + \frac{60}{343} + \frac{80}{343} \\ \\ = \frac{224}{343}$
Hope it helps you