The screens used for a certain type of cell phone are manufactured by 3 companies, A, B, and C. The proportions of screens supplied by A, B, and C are 0.5, 0.3, and 0.2, respectively, and their screens are defective with probabilities 0.01, 0.02, and 0.03, respectively. Given that the screen on such a phone is defective, what is the probability that Company A manufactured it?

The probability is 0.2941

Step-by-step explanation:

Let's call A the event that company A manufactured the screen, B the event that company B manufactured the screen, C the event that the company C manufactured the screen and D the event that the screen is defective.

The probability P (A/D) that the company A manufactures the screen given that is defective is:

P (A/D) = P (A∩D) / P (D)

Where P (D) = P (A∩D) + P (B∩D) + P (C∩D)

So, the probability P (A∩D) that the screen is manufactured by company A and the screen is defective is calculated as:

P (A∩D) = 0.5 * 0.01 = 0.005

Where 0.5 is the proportion of screens manufactured by company A and 0.01 is the probability of defective screens for screens that are manufactured by company A.

At the same way, the probability P (B∩D) that the screen is manufactured by company B and the screen is defective and the probability P (C∩D) that the screen is manufactured by company B and the screen is defective are calculated as:

P (B∩D) = 0.3 * 0.02 = 0.006

P (C∩D) = 0.2 * 0.03 = 0.006

Then, The probability P (D) that the screen is defective is equal to:

P (D) = 0.005 + 0.006 + 0.006 = 0.017

Finally, P (A/D) that the company A manufactures the screen given that is defective is:

P (A/D) = 0.005/0.017 = 0.2941