Sheldon Ross Example 3.7f

Question

A plane is missing and it is presumed that it was equally likely to have
gone down in any of three possible regions. Let 1 − αi denote the probability the plane
will be found upon a search of the ith region when the plane is, in fact, in that region,
i = 1, 2, 3. (The constants αi are called overlook probabilities because they represent the
probability of overlooking the plane; they are generally attributable to the geographical
and environmental conditions of the regions.) What is the conditional probability that the
plane is in the ith region, given that a search of region 1 is unsuccessful, i = 1, 2, 3?

Comments

ans 1/9?

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Value for alpha1 or anything is not given so the answer is expected with a term in alpha1.

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we can calculate alpha

rt?

Answer 1

Plane has equally likely probability to found in i th region

So, probability should be $\frac{1}{3}$

probability the plane will be found upon a search of the ith region ,

i.e. for 1st region $\left ( 1-\alpha .1 \right )=\left ( 1-\alpha \right )$

for 2nd region $\left ( 1-\alpha .2 \right )=\left ( 1-2\alpha \right )$

for 3rd  region $\left ( 1-\alpha .3 \right )=\left ( 1-3\alpha \right )$

Total probability should be 1

$\left ( 1-\alpha \right )+\left ( 1-2\alpha \right )+\left ( 1-3\alpha \right )=1$

$\alpha =\frac{1}{3}$

So, as given that a search of region 1 is unsuccessful

and probability is equally likely =$\frac{1}{3}\left ( 1-\frac{2}{3} \right )+\frac{1}{3}\left ( 1-\frac{3}{3} \right )=\frac{1}{9}$

Comments

i.e not alpha * i ...  i.e alpha subscript i....  i.e alpha1, alpha2......

Answer 2

P(1|f1^(c)) = alpha1/ (alpha1+2)

P(2|f1^(c)) = 1/(alpha1+2)

P(3|f1^(c)) = 1/(alpha1+2)

where f1^(c) represent the  search of region1 is unsuccessful