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3.4 Conditional Probability
In many cases, a probability depends on what we already know. For instance, would we believe
that the likelihood of a car accident changes, provided that the roads are slick from snow We
would probably agree that the likelihood increases if we already know the road conditions.
Suppose a fair, two-sided coin is tossed. You are told that the outcome is not a head. What is the
likelihood that the outcome is tails
The answer is probably obvious… if you know the outcome was not heads, and the only two
possibilities are heads and tails, then there is a 100% chance the outcome is tails.
This is a conditional probability. That is, if
Further, to indicate that the outcome is not one of the above, we often put a bar on top of the
event name:
Then,
̅
̅
( )
However, given that we know the outcome was not tails, the probability of heads jumped to 1.
We might write:
( ̅)
Instead of using the word “given” we often use a vertical line (called a “pipe”), |. That is,
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