Events and outcomes
01
Sample space - roll a die, flip a coin
A normal 6-sided die is cast, and then a coin is flipped. All results are recorded.
- (a) Define a sample space for this experiment.
- (b) How many possible events are there?
02
Sample space - roll a die then flip coin(s)
A normal 6-sided die is cast. If the result is even, flip a coin two times; if the result is odd, flip a coin one time. All results are recorded.
- (a) Define a sample space for this experiment.
- (b) How many possible events are there?
03
Events - descriptions to sets
You are modeling quality assurance for cars coming off an assembly line. They are either good (G) or broken (B). You watch 4 cars come off and record their status as a sequence of these letters, for example ‘GGBG’.
Determine the sets defined by the events having the following descriptions:
- (a) “third car is broken”
- (b) “all cars have the same status”
- (c) “at least one car is broken”
- (d) “no consecutive cars have the same status”
Probability models
04
Venn diagrams - set rules and Kolmogorov additivity
Suppose we know three probabilities of events:
, , and . Calculate:
, , , , and .
05
Inclusion-exclusion reasoning
Your friend says: “according to my calculations, the probability of
is and the probability of is , but the probability of and both happening is only .” You tell your friend they don’t understand probability. Why?
06
Inclusion-exclusion reasoning
Suppose
and . Show that .
07
At least two heads from three flips
A coin is flipped three times.
What is the probability that at least two heads appear?
Conditional probability
08
Conditioning - restrict to 4th-year students
Student test-passing rates, by year:
1st year 2nd year 3rd year 4th year Pass 0.155 0.340 0.255 0.160 Fail 0.025 0.040 0.015 0.010 What is the likelihood that a randomly chosen 4th-year student passed the test? What about for 1st-year students?
09
Conditioning - two dice, at least one is 5
Two dice are rolled, and at least one is a 5.
What is the probability that their sum is 10?
10
Conditioning - two dice, differing numbers
Two dice are rolled, and the outcomes are different.
What is the probability of getting at least one 1?
11
Multiplication - drawing two hearts
Two cards are drawn from a standard deck (without replacement).
(a) What is the probability that both are hearts?
(b) What is the probability that both are 4?