Conditional Probability
Example 1:
A box has 5 computer chips. Two are defective. Two chips are selected from the box, one at a time.
• Compute the probability that the second chip is defective.
Ω
=
{(g1 , g2 ), (g1 , g3 ), (g2 , g3 ), (g2 , g1 ), (g3 , g1 ), (g3 , g2 ), (g1 , d1 ), (g1 , d2 ), (g2 , d1 ), (g2 , d2 )
(g3 , d1 ), (g3 , d2 ), (d1 , g1 ), (d1 , g2 ), (d1 , g3 ), (d2 , g1 ), (d2 , g2 ), (d2 , g3 ), (d1 , d2 ), (d2 , d1 )}
Event A
= the second chip is defective
A
= {(g1 , d1 ), (g1 , d2 ), (g2 , d1 ), (g2 , d2 ), (g3 , d1 ), (g3 , d2 ), (d1 , d2 ), (d2 , d1 )}
P ( second chip is defective) =
8
|A|
=
= .4.
|Ω|
20
• If we know that the first chip is good, what is the probability that the second chip is defective.
The sample space for drawing two chips where the first chip good is, Ω1 , where
Ω1 = {(g1 , g2 ), (g1 , g3 ), (g2 , g3 ), (g2 , g1 ), (g3 , g1 ), (g3 , g2 ), (g1 , d1 ), (g1 , d2 ), (g2 , d1 ), (g2 , d2 ), (g3 , d1 ), (g3 , d2 )}
Event A1
=
first chip is good and the second chip is defective
A1
=
{(g1 , d1 ), (g1 , d2 ), (g2 , d1 ), (g2 , d2 ), (g3 , d1 ), (g3 , d2 )}
P (second chip is defective given first chip is good ) =
|A1 |
6
=
= .5.
|Ω1 |
12
Example 2:
Re-compute the probability that the second chip is defective given that the first chip is good using the definition
of conditional probability
First define event B in sample space Ω
Event B
B
= first chip is good
= {(g1 , g2 ), (g1 , g3 ), (g2 , g3 ), (g2 , g1 ), (g3 , g1 ), (g3 , g2 ), (g1 , d1 ), (g1 , d2 ), (g2 , d1 ), (g2 , d2 )
(g3 , d1 ), (g3 , d2 )}
Thus
P (B)
12
|B|
=
|Ω|
20
=
The event A ∩ B in sample space Ω is
Event A ∩ B = first chip is good and the second chip is defective
A = {(g1 , d1 ), (g1 , d2 ), (g2 , d1 ), (g2 , d2 ), (g3 , d1 ), (g3 , d2 )}
Thus
P (A ∩ B)
=
6
|A ∩ B|
=
|Ω|
20
Hence, using the definition of conditional probability
P (A|B)
=
=
P (A ∩ B)
P (B)
6/20
= .5.
12/20
Example 3:
More computer chips...A box has 500 computer chips with a speed of 400 Mhz and 500 computer chips with a
speed of 500 Mhz. The numbers of good (G) and defective (D) chips at the two different speeds are as shown in
the table below.
G
D
400 Mhz
480
20
500
500 Mhz
490
10
500
970
30
Total=1000
We select a chip at random and observe its speed. What is the probability that the chip is defective given that
its speed is 400 Mhz?
We know that P (D) = .03,
P (400 Mhz) = .5,
P (D ∩ 400 Mhz) = .02
Thus
P (D|400 Mhz) =
P (D ∩ 400Mhz)
.02
=
= .04
P (400 Mhz)
.5
Example 4:
Consider three cards. One card has two green sides, one card has two red sides, and the third card has one green
side and one red side.
- I pick a card at random and show you a randomly selected side: Ω = {{G, G}, {R, R}, {R, G}}
- What is the proability that the flip side is green given that the side I show you is green?
Example 5:
An alternative model for logging on to the AOL network using dial-up.
Suppose I log on to AOL using dial-up. I connect successfully if and only if the phone number works and the
AOL network works. The probability that the phone works is .9, and the probability that the network works is
.6. Suppose that the status of the phone line and the status of the AOL network are independent. What is the
probability that I connect successfully?
Example 6:
Events A and B are independent. Then
• A and B are independent
• A and B are independent
• A and B are independent.