Due date: Sunday 8/31, 11:59pm
Events and outcomes
01
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?
Link to originalSolution
01
(a)
(1) Count outcomes:
Since there are 6 possible results of rolling a die, and
possible results of a coin flip, our sample space has 12 elements.
(2) Use set builder notation to describe the sample space
:
(b)
Note that the number of possible events amount to counting how many subsets there are of
. In other words, we are asked to compute . Compute
. Link to original
02
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?
Link to originalSolution
02
(a)
(1) Divide the sample space into two disjoint sets:
Denote
as the sample space where the result of the die is even. Denote
as the sample space where the result of the die is odd.
.
(2) Describe
. There are
even numbers on a die, and possible results of each coin flip. Since coin flips are independent has elements.
(3) Write
using set builder notation.
(4) Describe
. There are
odd numbers on a die, and 2 possible results of a coin flip. has elements.
(5) Write
using set builder notation.
(6) Describe
. As above,
. Since
and are disjoint, .
(b)
(1) Note that the number of possible events amount to counting how many subsets there are of
. In other words, we are asked to compute .
(2) Compute
. Link to original
Probability models
03
01
Venn diagrams - set rules and Kolmogorov additivity
Suppose we know three probabilities of events:
, , and . Calculate:
, , , , and . Link to originalSolution
04
(1)
is computed by directly applying the inclusion-exclusion principle.
(2)
(3)
(4) We can express
as . Therefore,
(5)
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04
04
At least two heads from three flips
A coin is flipped three times.
What is the probability that at least two heads appear?
Link to originalSolution
07
(1) Describe the sample size of this experiment.
(2) Find the probability that at least two heads appear.
The sequences of flips that contain at least two heads are
, , , . We know that
, thus Link to original
Conditional probability
05
01
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?
Link to originalSolution
08
(1) We are asked to compute
. Set up the conditional probability formula.
(2) We have from the table that
and . Therefore,
(3) For 1st-year students, we have
Link to original