Question

Box 1 contains 30 cards marked from 1 to 30 and what to contain 20 cards marked from 31 to 50. A box is selected and a card is drawn. If the number on the card is non-prime then what is the probability that it came from Box 1?
4/17
6/17
7/17
8/17 ​

Answer 1

Box 1 contains 30 cards marked from 1 to 30 and what to contain 20 cards marked from 31 to 50. A box is selected and a card is drawn.

To find : If the number on the card is non-prime then what is the probability that it came from Box 1.

solution : there are two boxes, right ?

if we select one out of two, probability of selecting each box = 1/2

i.e., P(B₁) = P(B₂) = 1/2

non prime numbers between 1 to 30 = 1 , 4 , 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22 , 24 , 25, 26, 27, 28, 30 = 20 integers

so, probability to get non prime numbers, P(B'₁) = 20/30 = 2/3

non prime numbers between 31 to 50 = 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50 = 15 integers

so probability to get non prime numbers , P(B'₂) = 15/20 = 3/4

non probability of non prime that it came from box 1 =P(B₁)P(B'₁)/[ P(B₁)P(B'₁) + P(B₂)P(B'₂)]

= (1/2 × 2/3)/(1/2 × 2/3 + 1/2 × 3/4)

= (1/3)/(1/3 + 3/8)

= (1/3)/(17/24)

= 8/17

Therefore the probability that it came from box 1 is 8/17

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

Let A & B be the boxes

$\displaystyle \: P(A) = P(B) = \frac{1}{2}$

Then

A contains 30 cards marked from 1 to 30

B contains 20 cards marked from 31 to 50 Let X be the event that the card is non-prime

Now the prime numbers from 1 to 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Again the prime numbers from 31 to 50 is 31, 37, 41 , 43, 47

Now the number non prime numbers from 1 to 30 is 20 & the number non prime numbers from 31 to 50 is 15

So$\displaystyle \: P( X/A) = \frac{20}{30} = \frac{2}{3}$

$\displaystyle \: P(X/B) = \frac{15}{20} = \frac{3}{4}$

So the required probability is

$\displaystyle \: P(A/X)$

$\displaystyle \: = \frac{P(A) \times \: P(X/A) }{P(A) \times \: P(X/A) +P(B) P(X/B) }$

$\displaystyle \: = \frac{ \frac{1}{2} \times \frac{2}{3} }{ \frac{1}{2} \times \frac{2}{3} + \frac{1}{2} \times \frac{3}{4} }$

$\displaystyle \: = \frac{ \frac{2}{3} }{ \frac{2}{3} + \frac{3}{4} }$

$\displaystyle \: = \frac{ \frac{2}{3} }{ \frac{17}{12} }$

$\displaystyle \: = \frac{2}{3} \times \frac{12}{17}$$\displaystyle \: = \frac{8}{17}$