GATE CSE 2005 | Question: 51

Question

Box $P$ has $2$ red balls and $3$ blue balls and box $Q$ has $3$ red balls and $1$ blue ball. A ball is selected as follows: (i) select a box (ii) choose a ball from the selected box such that each ball in the box is equally likely to be chosen. The probabilities of selecting boxes $P$ and $Q$ are $\dfrac{1}{3}$ and $\dfrac{2}{3}$ respectively. Given that a ball selected in the above process is a red ball, the probability that it came from the box $P$ is:

• A. $\dfrac{4}{19}$

• B. $\dfrac{5}{19}$

• C. $\dfrac{2}{9}$

• D. $\dfrac{19}{30}$

Answer 1

The probability of selecting a red ball,

$=\left(\dfrac{1}{3}\right)* \left(\dfrac{2}{5}\right)+ \left(\dfrac{2}{3}\right)* \left(\dfrac{3}{4}\right)$

$ = \dfrac{2}{15} + \dfrac{1}{2}= \dfrac{19}{30}$

Probability of selecting a red ball from box,

$P=\left(\dfrac{1}{3}\right)* \left(\dfrac{2}{5}\right) = \dfrac{2}{15}$

Given that a ball selected in the above process is
a red ball, the probability that it came from the
box $P$ is $ =\left({\dfrac{2}{15} }\div { \dfrac{19}{30}}\right)= \dfrac{4}{19}$

Correct Answer: $A$

Comments

But it says each ball is equally likely to be chosen. Probabilities of red ball should be 1/5 and 1/4

Answer 2

Answer : Option A 

Comments

It is Bayes' formula... he sort of even derived it

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@Devwritt @Shaik Masthan @Mk Utkarsh

I use this method only to solve this type of problem, but pls clear 1 doubt.

for P(A intersection B) we write P(A).P(B)   {in case of independent events}  or P(A).P(B/A) {in case of dependent events}. But how to check which one to use...( i.e. how to determine whether the case is of dependent or independent ) ? 

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Normally they given, otherwise we have to infer from data

is P and Q are independent or not ?

if P is rolling a die and Q is tossing a coin, then Find the probability of ( head with die is grater than 2)

in this case P and Q are independent ==> P(P $\cap$ Q) = P(P).P(Q)

Answer 3

Given that a ball selected in the above process is a red ball, the
probability that it came from the box P is:

P(comes from box p / selected ball is red) 

= P(comes from box p ∩ selected ball is red) / p (selected ball is red)

here P(comes from box p ∩ selected ball is red) = (1/3)*(2/5) = 2/15

p (selected ball is red) = (1/3)*(2/5) + (2/3) * (3/4) = 19/30

so required probility is = (2/15) / (19/30) = 4/19

Comments

THIS IS MADE EASY METHOD

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It is also the bayes theorem itself.

Answer 4

Now Applying Baye's Theorem Directly:-

P(P/Red) =[ P(P)* P(Red/P) ] / [ P(P)* P(Red/P)  + P(Q)* P(Red/Q) ]

= ( 1/3 * 2/5 ) / [ (1/3 * 2/5) + (2/3 * 3/4) ]

= 4/19