Total Probability and Bayes' Theorem, Quiz

Bag I has

3

red and

5

white balls; bag II has

6

red and

2

white. A bag is chosen at random and a ball drawn. Find

P(\text{red})

.

A

\displaystyle \tfrac{1}{2}

B

\displaystyle \tfrac{9}{16}

C

\displaystyle \tfrac{3}{8}

D

\displaystyle \tfrac{9}{8}

In the previous situation (bag I:

3

red,

5

white; bag II:

6

red,

2

white), the ball drawn is red. Find the probability that it came from bag II.

A

\displaystyle \tfrac{1}{2}

B

\displaystyle \tfrac{3}{4}

C

\displaystyle \tfrac{2}{3}

D

\displaystyle \tfrac{1}{3}

Two machines make

60\%

and

40\%

of a factory's output, with defect rates

1\%

and

2\%

. Find the probability that an item picked at random is defective.

A

0.014

B

0.015

C

0.03

D

0.008

In the previous situation (shares

60\%

,

40\%

; defect rates

1\%

,

2\%

), an item is found defective. Find the probability that the second machine made it.

A

\displaystyle \tfrac{2}{3}

B

\displaystyle \tfrac{4}{7}

C

\displaystyle \tfrac{3}{7}

D

0.4

In Bayes' theorem, the probabilities

P(E_i \mid A)

, found after observing

A

, are called

A total probabilities

B independent probabilities

C prior probabilities

D posterior probabilities

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