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?
Solution
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
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?
Solution
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
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”
Solution
03
(a) Since only the third car is broken, and the other three cars can have any status, the relevant set
is (b) In this case, either all cars are good or all cars are broken. Therefore, the relevant set
is (c) In this scenario, the only combination not in the relevant set is ‘GGGG’. Therefore,
(d) In this scenario, given two cars
and , . Therefore, Link to original