Transcript 4.6 Conditional Probability

3.5 Conditional Probability
Conditional Probability
• Dependent events
– When the outcome of a second event depends
directly on the outcome of the first event
• Conditional Probability
– The probability that B occurs, given A has
already occurred
– We write P(B | A)
– We say “the probability of B given A”
Example
• A professional hockey team has eight wingers.
Three of these wingers are 30-goal scorers, or
“snipers.” Every fall, the team plays an exhibition
match with the club’s farm team. In order to make
the match more interesting for the fans, the
coaches agree to select two wingers at random
from the pro team to play for the farm team. What
is the probability that two snipers will play for the
farm team?
Solution
•
•
•
•
A = {first winger is a sniper}
B = {second winger is a sniper}
Are these independent or dependent events?
Dependent: once you select a sniper, you
don’t put him back in to the pot
P( A) 
P( B | A) 
3
8
2
7
P( A  B)  P( A) P( B | A)
 3  2  1
   
4 8  7 
3

28
Conditional Probability
• If A and B are dependent events
P(A  B) = P(A)P(B | A)
For independent events, P(B | A) = P(B)
Conditional Probability:
P( A  B)
P( B | A) 
P( A)
Example
Shy Tenzin’s friends assure him that if he asks
Mikala out on a date, there is an 85%
chance that she will say yes. If there is a
60% chance that Tenzin will summon the
courage to ask Mikala out to the dance next
week, what is the probability that they will
be seen at the dance together?
Solution
• A = {Tenzin summons courage to ask
Mikala out}
P(A) = 0.60
• B = {Mikala says yes}
• Note that B is conditional on Tenzin getting
the courage, that is, B depends on A
P(B | A) = 0.85
There is a 0.51
P(A  B) = P(A)P(B | A)
probability that Tenzin
= (0.60)(0.85)
and Mikala will be at
= 0.51
the dance together.