MATH 180A Lecture 4

Section 1.4

Decompositions

In this section we will see many important uses of the identity for disjoint events

Example. A fair coin is flipped 5 times. Calculate the probability that it lands tails at least 3 times.

” T”
” T”
” T”
” T”
are disjoint,
and
means

# = choosing 3 positions out of 5 to be T =

Complements

Fact. Let $A$ be an event in a probability space and $A^C=\Omega \setminus A$ its complement.

Then

Proof.
(disjoint)

Example. 4 fair dice are rolled. Calculate the probability that we get at least one pair of doubles.

look at complement
”at least one pair of doubles"
"don’t get at least 1 pair of doubles"
"4 rolls are distinct”

Birthday Problem:
“at least 2 same bday""distinct bdays”

Inclusion-Exclusion

Example. In a given country, 20% of the population owns a cat, 25% owns a dog, and 5% owns one of each.
What is the probability that a person chosen uniformly at random from this country owns neither a cat nor a dog?

= person owns a cat
= person owns a dog
owns neither = de Morgan’s Law

Side Note de Morgan's Law

Also says

“owns neither”

Monotonicity

Fact. If $A\subseteq B$, then $\mathbb{P}(A)\le \mathbb{P}(B)$

Proof.



.

Example. Suppose that out of the total North American population, 50% of the
people have at some point in their lives visited the US, 30% has visited Texas, and 40% has visited California. Knowing only this information, what is the smallest possible percentage of the North American population that has visited both Texas and California? And largest?

= “person visited Texas”
= “person visited California”
= “person visited US”

Smallest is 20%

b/c and

30% is largest

Section 2.1

Conditional Probability

Example.

Your friend rolls a pair of fair dice. What is the probability that the sum is 10?

“sum is 10”

Your friend rolls a pair of fair dice and tells you that the sum is a 2 digit number. Now what is the probability that the sum is 10?

Your friend rolls a pair of fair dice and tells you that one of the rolls is a 6. What is the probability that the sum is 10?

answer not