MATH 180A Lecture 3

Source: https://canvas.ucsd.edu/courses/55106/pages/lecture-notes-and-videos

Example: w/ replacement, order matters: I flip a fair coin 6 times.

Describe the sample space

Calculate $\mathbb{P}(\text{every even flip is heads})$

Example: w/o replacement, order matters

There are 6 numbered balls in an urn. 3 are removed w/o replacement and lined up in order.

Calculate the probability that the 1st tow numbers removed are (3,6) (in that order).

Suppose 5 balls are removed instead of 3. Now what is the probability that the 1st two numbers removed are (3,6)?

An urn contains 10 marbles: 5 are red, 2 are white, 3 are blue. Three are chosen uniformly at random and taken out.

Calculate the probability that we drew 2 blue and 1 red.

Instead, suppose that the urn contained 6 red, 4 white, and 7 blue marbles and that we drew three uniformly at random. Now calculate the probability that we drew 2 blue and 1 red.

Section 1.3

Infinitely Many Outcomes

Example. A number is chosen “uniformly at random” from the interval .

Describe the probability space modeling this experiment

subset
all subsets of [0,2] which “can be reasonably assigned a length”
e.g. length([a,b]) = b-a
length([1/3,1/2]U[1,1.3]))=(1/2-1/3)+(1.3-1)

What is $\mathbb{P}(X\ge 0.5)$?

What is $\mathbb{P}(X=1)$?

doesn’t mean X=a is impossible
because [0,2] is uncountable

Example. An archery target is a 50cm diameter disk containing a middle disk of 25cm and a bullseye of 5cm. You shoot an arrow which hits a point uniformly at random on the target.

Describe the probability space modeling this experiment

all subsets of which we can “reasonably assign an area”

Calculate the probability that the arrow hits the middle disk but misses the bullseye.

Calculate the probability that the arrow hits the bullseye.