Question

Two identical boxes have chips in them. Box I has 4 blue chips and 2 red chips. Box II has 2 blue chips and 6 red chips. A box is randomly selected, and one chip is randomly drawn.

a) What is the probability of drawing a red chip?

b) Given that Box I was chosen, what is the probability of drawing a red chip?

c) Based on your answers in parts a & b, are the events “drawing 1 red chip” and “choosing Box I” dependent or independent? Justify your answer.

Answer 1

Solution:

Given:

Box I:  4 blue chips and 2 red chips. Total = 6 chips

Box II : 2 blue chips and 6 red chips. Total = 8 chips

A box is randomly selected, and one chip is randomly drawn.

Thus

P(Box I )= 0.5 and P(Box II)=0.5 ( since both box has same chance of selection)

Let BI =Box I and BII = Box II

then

P(BI) = 0.5 and P(BII) = 0.5

Part a) What is the probability of drawing a red chip?

P(Red) =...........?

Let R = Red chip

thus we have to find P(R) = .........?

$P(R)=P(R|BI) \times P(BI) +P(R|BII) \times P(BII)$

$P(R)= \frac{2}{6} \times 0.5 + \frac{6}{8} \times 0.5$

$P(R)= 0.3333333 \times 0.5 + 0.75 \times 0.5$

$P(R)=0.54166667$

$P(R)=0.541667$

Part b) Given that Box I was chosen, what is the probability of drawing a red chip?

P(R|BI) =.............?

$P(R|BI) = \frac{2}{6}$

Part c)

Since  $P(R|BI) =0.333333 \neq P(R)=0.541667$

Thus  the events “drawing 1 red chip” and “choosing Box I” dependent.