### Probability Rules

```Probability Rules!


When two events A and B are disjoint, we have
P (A or B) = P (A) + P (B)
However, when the events are not disjoint, this
addition rule will double count the probability
of both A and B occurring. Thus, we need the
P (A or B) = P (A) + P (B) – P (A and B)

The following Venn diagram
shows a situation in which we
would use the general addition rule:




A check of dorm rooms on a college campus
refrigerators, and 43% had both a TV and a
refrigerator. What’s the probability that a
randomly selected dorm room has
a refrigerator but no TV?
either a refrigerator or a TV?
neither refrigerator nor TV?



When we want the probability of an event from
a conditional distribution, we write P (B|A) and
pronounce it “the probability of B given A.”
A probability that takes into account a given
condition is called a conditional probability.
To find the probability of the event B given the
event A, we restrict our attention to the
outcomes in A. We then find in what fraction of
those outcomes B also occurred.
P(B| A)  P(A and B)
P(A)





Draw a card at random from a deck of 52
cards. What’s the probability that
the card is a spade, given that it is black?
the card is a king, given that it is red?
the card is a king, given that is a face card?
the card is red, given that it is a diamond?


Independence of two events means that the
outcome of one event does not influence the
probability of the other.
Formally, events A and B are independent
whenever P (B|A) = P (B) , or equivalently,
whenever P (A|B) = P (A).

Disjoint events cannot be independent:
◦ Since we know that disjoint events have no
outcomes in common, knowing that one occurred
means the other didn’t.
◦ Thus, the probability of the second occurring
changed based on our knowledge that the first
occurred.
◦ It follows, then, that the two events are not
independent.


When two events A and B are independent,
P (A and B) = P (A) x P (B)
However, when our events are not independent,
the above rule does not work. Thus, we need the
General Multiplication Rule:
P (A and B) = P (A) x P (B|A)
or
P (A and B) = P (B) x P (A|B)


Sampling without replacement means that once
one object is drawn it doesn’t go back into the
pool.
◦ We often sample without replacement, which
doesn’t matter too much when we are dealing
with a large population.
◦ However, when drawing from a small population,
we need to take note and adjust probabilities
accordingly.
Drawing without replacement is just another
instance of working with conditional probabilities.




Draw two cards at random from a deck of 52
cards. What’s the probability that
both cards are black?
at least one card is red?
the first card is an ace?


A tree diagram helps us think through
conditional probabilities by showing
sequences of events as paths that look
like branches of a tree.
Examples:
◦ Toss a coin 3 times.
◦ Suppose a person aged 20 has about an 80%
chance of being alive at age 65. Suppose that
three people aged 20 are selected at random.
What’s the probability that exactly two people
will be alive at age 65?
Solution


Suppose we want to know P (A|B), but we know only
P (A), P (B), and P (B|A). We also know P (A and B),
since P (A and B) = P (A) x P (B|A)
From this information, we can find P (A|B):
P(A|B)  P(A and B)
P(B)

When we reverse the probability from the
conditional probability that you’re originally given,
We obtain Bayes’s Rule.
P A | B P B 
P B | A  
C
C
P A | B P B   P A | B P B

 


Page 404 – 407
Problem # 1, 3, 5, 7, 9, 11, 15, 17, 19, 23,
27, 29, 33, 35, 43, 45.
```