Question

An experiment consists of choosing an urn with the following probabilities that Urn 1, Urn 2, or Urn 3 will be chosen: 1/2, 1/4, and 1/4, respectively. Urn 1 contains 15 brown marbles and 14 clear marbles. Urn 2 contains 6 brown marbles, 11 clear marbles and 9 red marbles. Urn 3 contains 7 brown marbles, 8 clear marbles and 5 red marbles.
A marble is then chosen from the chosen urn. What is the probability that Urn 3 was chosen, given that the marble chosen was clear?

a) 0.1000

b) 0.4000

c) 0.1333

d) 0.3063

e) 0.2236

Answer 1

Answer:

option E

Step-by-step explanation:

let E₁ be the event of urn 1

let E₂ be the event of urn 2

let E₃ be the event of urn 3

P(E₁) = $\dfrac{1}{2}$

P(E₂) = $\dfrac{1}{4}$

P(E₃) = $\dfrac{1}{4}$

Let A be the probability of getting clear marble

P(A/E₁) = $\dfrac{14}{29}$ =0.483

P(A/E₂) = $\dfrac{11}{26}$ = 0.423

P(A/E₃) = $\dfrac{8}{20}$ =  0.4

required probability

P(E₃/A) = $\dfrac{P(E_3)P(A/E_3)}{P(A)}$

P(A) = P(E₁)P(A/E₁) + P(E₂)P(A/E₂) +P(E₃)P(A/E₃)

P(A) = 0.5 x 0.483  + 0.25 x 0.423 +0.25 x 0.4

P(A) = 0.447

now,

P(E₃/A) = $\dfrac{0.25 \times 0.4}{0.447}$

P(E₃/A) = 0.2147

hence, the correct answer is option E