7. Consider a box that contains 10 red balls and 8 blue balls. A person draws a

Question

7. Consider a box that contains 10 red balls and 8 blue balls. A person draws a ball from the box and shows it to their partner at some distance from the site. Unfortunately, the partner is viewing the balls over a fuzzy environment with the result that if the ball drawn is a red ball, they will say that it is a red ball with probability 0.8 and that it is a blue ball with probability 0.2. Similarly, if the ball drawn is a blue ball they will say that it is a red ball with probability 0.3 and that it is blue ball with probability 0.7. The balls are drawn with replacement. a. Given that the partner says that the ball drawn is a red ball, what is the probability b. Given that the partner says that the ball drawn is a blue ball, what is the What is the probability that the partner makes a mistake in identifying an arbitrary that it is actually a red ball? probability that it is actually a red ball? ball? c.

Explanation / Answer

Here we have 10 red balls and 8 bue balls. Therefore P( Red actually ) = 10/18 = 0.5556

P( blue actually ) = 8/18 = 0.4444

Also we are given that:

P( partner reads red | red actually ) = 0.8 and P( partner reads blue | red actually ) = 0.2

P( partner reads red | blue actually ) = 0.3 and P( partner reads blue | blue actually ) = 0.7

Therefore by law of total probability we get:

P( partner reads red ) = P( partner reads red | red actually ) P( Red actually ) + P( partner reads red | blue actually )P( blue actually )

P( partner reads red ) = 0.8*(10/18) + 0.3*(8/18) = 0.5778

Now Using bayes theorem, we have:

P( red actually | partner reads red ) P( partner reads red ) = P( partner reads red | red actually ) P( Red actually )

Therefore, we get:

P( red actually | partner reads red ) = P( partner reads red | red actually ) P( Red actually ) / P( partner reads red )

P( red actually | partner reads red ) = 0.8*(10/18) / 0.5778 = 0.7692

Therefore 0.7692 is the required probability here.

b) Now as we found out that P( partner reads red ) = 0.5778, therefore

P( partner reads blue ) = 1- 0.5778 = 0.4222

By bayes theorem, we get:

P( red actually | partner reads blue ) P( partner reads blue ) = P( partner reads blue | red actually ) P( red actually )

P( red actually | partner reads blue ) = P( partner reads blue | red actually ) P( red actually ) / P( partner reads blue )

P( red actually | partner reads blue ) = 0.2*(10/18) / 0.4222 = 0.2632

Therefore 0.2632 is the required probability here.

c) Probability that partner makes a mistake in identifying an arbitrary ball is computed as:

= P( partner reads blue | red actually ) P( red actually ) + P( partner reads red | blue actually ) P( blue actually )

= 0.2*(10/18) + 0.3*(8/18)

= 0.2444

Therefore 0.2444 is the required probability here.

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