Question

A boy has color blindness and has trouble distinguishing blue and green. There are 75 blue pens and 25 green pens mixed together in a box. Given that he picks up a blue pen, there is a 80% chance that he thinks it is a blue pen and a 20% chance that he thinks it is a green pen. Given that he picks a green pen, there is an 90% chance that he thinks it is a green pen and a 10% chance that he thinks it is a blue pen. Assume that the boy randomly selects one of the pens from the box.
a) (3pts) What is the probability that he picks up a blue pen and recognizes it as blue?



b) (3 pts) What is the probability that he chooses a pen and thinks it is blue?



c) (4pts) Given that he thinks he chose a blue pen, what is the probability that he actually chose a blue pen?

Answer 1

Answer:

a) 60% probability that he picks up a blue pen and recognizes it as blue.

b) 62.5% probability that he chooses a pen and thinks it is blue.

c) 96% probability that he actually chose a blue pen.

Step-by-step explanation:

The problem has the following probabilities:

75 of 100 pens are blue, so a 75% probability that a pen is blue.

25 0f 100 pens are green, so a 25% probability that a pen is green.

If a pen is blue, an 80% probability that he thinks it is a blue pen

If a pen is blue, a 20% probability that he thinks it is a green pen.

If a pen is green, a 90% probability that he thinks it is a green pen.

If a pen is green, a 10% probability that that he thinks it is a blue pen.

a) (3pts) What is the probability that he picks up a blue pen and recognizes it as blue?

75% probability he picks up a blue pen.

80% that he recognizes a blue pen as blue.

$P = 0.75*0.8 = 0.6$

60% probability that he picks up a blue pen and recognizes it as blue.

b) (3 pts) What is the probability that he chooses a pen and thinks it is blue?

75% probability that he picks up a blue pen.

80% that he recognizes a blue pen as blue.

25% probability that he picks up a green pen

10% probability that he thinks a green pen is blue.

$P = 0.75*0.80 + 0.25*0.10 = 0.625$

62.5% probability that he chooses a pen and thinks it is blue.

c) (4pts) Given that he thinks he chose a blue pen, what is the probability that he actually chose a blue pen?

62.5% probability that he thinks that he choose a blue pen.

60% probability that he chooses a blue pen and think that it is blue.

$P = \frac{0.6}{0.625} = 0.96$

96% probability that he actually chose a blue pen.