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   Which is more likely – rolling 2 sixes in a row, or rolling
3? Why?

   Which is more likely – flipping heads twice in a row, or
rolling a six? Why do you think so?
Compound Probability
Wednesday 28 November 2007
This week…
•   You should be able to…

 Determine sample spaces
 Count outcomes using combinations and
permutations
 Determine probabilities of compound events

 Determine probabilities in ‘choosing’ problems
A note about ‘or’
   Were you at the mall, or the movies?
   Yep.
   What?
   Yep.

   In real life, ‘or’ is generally exclusive.
   In math, it’s usually inclusive.
   The difference determines whether or not this exchange
makes sense.
The Complement
   The complement is the
case where the event
doesn’t happen.

   It has probability 1 – p.
Unions
   The union of two events is
the sum of all cases where
one or the other takes place.

or
   Events are mutually exclusive if
they can’t both happen.
   Another term for this is
disjoint.

   Eg., Coming to school and
missing school are disjoint
events.
Unions of Events
   The union of 2 events is
the case where one, or
the other, or both
happens.

   Keyword:         OR

   Method:          +
Probabilities of Unions
   If A and B are exclusive, and
can’t both happen, then the
probability of the union is
the sum of the probabilities
of A and B.

   That is, when you’re talking
Probabilities of Unions
   If there’s a 20 % that a
randomly chosen car is
red, and a 14 % chance
it’s white…

   There’s a 34 % chance
that it’s red or white.
Probabilities of Unions
   If we simply add A and
B, we’re going to count
that green part twice.
   So, when we don’t know if events are distinct, the
formula is this:

P(A U B) = P(A) + P(B) - P(A  B)

=           +            -

That is, the probability of the union of A and B is equal to the
probability of A plus the probability of B minus the probability
of their intersection.
For Example
   64 % of homes have garages.
   21 % have swimming pools.
   17 % have both.

   What is the probability that a home has
 A pool or a garage?
 Neither a pool nor a garage?
 A pool but no garage?
Example 1
   A survey of college dorms found that 38 % had
refrigerators, 52 % had TVs, and 21 % had both.

   What is the probability that a randomly-chosen dorm
room will have:
   A TV but no refrigerator?
   A TV or a refrigerator, but not both?
   Neither a TV nor a refrigerator?
Example 2
   A survey of high school students found that 44 % had
computers, 21 % had mp3 players, and 14 % have both.

   What is the probability that a high school student has:
   One or the other, but not both?
   Neither an mp3 player nor a computer?
   Just an mp3 player?
   Just a computer?
   One, the other, or both?
Car Repairs
   Reports indicate that over a 1-year period, 17 % of cars will need
one repair, 7 % will need two repairs, and 4 % will need three or
more repairs.

   What is the probability that a car chosen at random will need…
   No repairs?
   No more than 1 repair?
   Some repairs?
   If you own two cars, what is the probability that…
   Neither will need repair?
   Both will need repair?
The Intersection of Two Events
 When  2 or more
things happen.

 Keyword:   AND

 Method:    X
The Intersection of Two Events
   A = “I roll a 4 on a number cube.”
   B = “I choose a club  from a deck
of cards.”
   A  B = “I roll a 4 and choose a
club.”
   If the two events are independent
P(A) = 0.16 and
P(B) = 0.25, then
P(A  B) = 0.16 X 0.25 = 0.04.
Car Repairs
   Reports indicate that over a 1-year period, 17 % of cars will need
one repair, 7 % will need two repairs, and 4 % will need three or
more repairs.

   What is the probability that a car chosen at random will need…
   No repairs?
   No more than 1 repair?
   Some repairs?
   If you own two cars, what is the probability that…
   Neither will need repair?
   Both will need repair?
M&M Colors
Color    Yellow     Red     Orange        Blue     Green    Brown

%         20       20        10           10       10       ???

   If you pick an M&M at                  If you pick three M&Ms in a
random, what is the                     row, what is the probability
probability that…                       that…
 It is brown?                           They are all brown?
 It is yellow or orange?                None are yellow?
 It is not green?                       At least one is green?
 It is striped?
Dice, again.
   You roll a fair die three times. What is the
probability that…
 You roll three 6’s?
 You roll all odd numbers?

 None of your rolls is divisible by 3?

 You roll at least one 5?

 You don’t roll all 5’s?
Summary
   Complement: doesn’t happen
   Method: 1 – p

   Union: one event or the
other
   Keyword: or

   Intersection: both events
   Keyword: and
   Method: multiplication

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