Task 1:
Question 1
Suppose we randomly draw a single student from the population of psychology students at UM.
Jim and Peter are two psychology students at UM. Which of the following statements is correct?
A Jim and Peter are two elementary outcomes of this random experiment
B Jim and Peter are two independent events of this random experiment
C The probability of drawing either Jim or Peter is given by the product: P (Jim) * P (Peter)
For A: Since we draw only a single student (N = 1), every unique student counts as an elementary
outcome. If you draw multiple times, this would not hold true however! For instance if we draw two
students, an elementary outcome might be “Jim, Peter” or “Peter, Peter” or “Jim, Jim”.
For B: An event is a subset of the sample space, made up of one or more Elementary Outcomes.
Events are statistically independent when one event doesn’t influence the probability of the other
event taking place. The question doesn’t state if there were any events defined- or how, so we
cannot assess whether this is true.
For C: This is the formula “P(A and B) = P(A) * P(B)”. This formula however only holds true in case
the events are independent. As we just said for question B; we cannot assess whether Jim and Peter
count as independent events. Had answer B been true however, then would answer C have been
correct? No! They ask “the probability of drawing either Jim or Peter”. That would be P(A or B),
while the formula given refers to P(A and B). Had B been true, the formula for question C would
have to look like P(A or B) = P(A|B) + P(B|A) – P(A|B) * P(B|A).
(You can derive this long formula by looking at other findings we had in case of independent events,
such as P(A) = P(A|B), which you can then fill in to the default formula for P(A or B))
Question 2
Consider the population of UM students. Suppose that 40% of all students at the faculty of
economics are female. Additionally, 65% of all students at the faculty of psychology are female.
Suppose that we draw a simple random sample (SRS) of size n=250 students from the population
of UM students. We record 'specialization' and 'gender' of each student in the sample. Which of
the following quantities is a random variable for this random experiment?
A the sample size
B the percentage of economy students at the UM
C the number of women in the sample
For A: The sample size is not a random variable, it is already provided that our sample size N =
250.
For B: The economy students at the UM are a population, they aren’t the sample. The percentage
of economy students at the UM is thus not a random variable, but a known and fixed amount.
For C: In the sample we assess the specialization (economy/psychology) and gender of each
randomly drawn student, so the number of women within the sample are a random variable. Could
be that you draw 250 students, and 145 of them turn out to be women, but if you repeat the study,
next time you might find 200 of the students in your sample are women.
Question 3
A psychology faculty has 1000 students, of which 400 are men and the other 600 are women.
Furthermore, 500 of the 1000 students are Dutch, 300 are Belgian and 200 are German. We
randomly draw a single person from this population. This is a random experiment. Which of the
following statements is incorrect?
A ‘the age of the student drawn is equal to 20’ is an elementary outcome of this random
experiment
B ‘the age of the student drawn is equal to 20’ is an event of this random experiment
C ‘the age of the student drawn’ is a random variable
,For A: An elementary outcome is each unique combination that you could draw in your sample.
Among your 1000 students, it’s unlikely that only a single one of them happens to be 20 years old.
So you could have drawn perhaps 400 different students for who this statement would hold true.
For B: An event is a subset of the sample space, made up out of one or more Elementary
Outcomes. Focussing specifically on all students who have the specific age of 20, would definitely
count as an event.
For C: A random variable is made up out of one or more events. As we stated B to be true, “age is
equal to 20” is an event which is one of the possible values for your random variable.
Question 4
Suppose we draw a SRS of size n=250 from the population of UM students. We record
'specialization' and 'gender' of each student in the sample. Suppose that 40% of students at the
faculty of economics are female. Additionally, 65% of all students at the faculty of psychology are
female. What is the sample space of this random experiment?
A All 250 students that were drawn
B All possible samples of size n=250 that can be possibly drawn from the population of UM
students
C All psychology and economy students together
The sample space is the set of all possible elementary outcomes. (E.g. If you flip a coin twice, your
sample space would be (H,T); (H,H); (T,H); (T,T).)
The 250 students that were drawn are the random sample (in this case the simple random sample).
All psychology and economy students together would be your complete population.
Question 5
Refer to the population described in question 4. We now consider the random experiment in
which we draw a single student from this population. Which of the following probabilities can be
obtained based on the given information?
A the conditional probability of a male student, given that he studies psychology
B the unconditional probability of an economy student
C the conditional probability of an economy student, given that she is a female
For A: This works perfectly fine with the provided info. Nothing is said about males, but we
already know the probability P(Female|Psychology). Namely: If we know the student studies
psychology, there is a 65% chance that this person is female. We could then use the complement
rule to find this probability for males. So:
P(Male|Psychology) = 1 – P(Female|Psychology) = 35%.
((Do note that this only holds true since we only have two options here (male and female). For
example, if we used genders instead of biological sex, and considered that nonbinary genders are
acknowledged in this day and age, all psychology students would have been made up out of (man,
woman, non-binary). Then using the complement rule as we did above would simply show “the
probability of a student who is not a woman, given that they study psychology”.))
For B: For the unconditional probability, you would need to know the total amount of economy
students, as well as the total amount of students. Neither are given, so you cannot assess this.
For C: For this to be possible, the info given should have been the other way around, such as
“Suppose that 40% of the students who are female, study at the faculty of economics. From that
info, if you already know the student is female, you could infer the probability that she also studies
economics.
Question 6
Suppose we are conducting a random experiment and that two events A and B are not disjoint.
This implies
A P (A and B) = 0
, B P (A or B) = P(A) + P(B)
C A and B may be two independent events
For A: This only applies in case of disjoint events.
For B: This only applies in case of disjoint events.
For C: If an event is “not” disjoint but no further info is provided, the only thing we know, is that the
formulas applicable to disjoint events, do not apply. However this info is not sufficient to determine
whether our event is then independent or dependent, as both independent and dependent events
can be joint. (Disjoint events are dependent events, but of the extreme kind, so any other
dependent event is still joint.) So if for instance an answer option had been “P(A) = P(A|B)” this
would not have been a valid answer, as this one occurs only in case of independent events, and we
cannot infer whether our event is dependent or independent from the info in the question. So
answer C is the only legitimate option here; A and B may be two independent events, but you
cannot say this with certainty.
Question 7
Consider the following contingency table in which 1000 persons are classified according to their
age (young vs old) and marital status (unmarried, married, divorced)
We randomly draw a single person from this population. Which of the following statements is
correct?
A the events ‘the person is married’ and ‘the person is young’ are independent
B the probability of drawing someone who is old and divorced is 0.67
C the conditional probability of drawing a young person, given that he or she is divorced is 0.33
For A:
P(Young) = = 0.55
P(Married) = = 0.4
If the events are independent, P(A) = P(A|B) so P(Young) = P(Young|Married)
P(Young|Married) = = 0.5
So they are not independent!
For B:
P(old and divorced) = = 0.1
For C:
P(Young | Divorced) = = 0.33
Question 8
A box contains 10 cards numbered 1,2,...,10. From this box a single card is drawn randomly. The
random variable X is the number of the drawn card. The expected value of X is equal to
A 5.5
B6
C This value cannot be determined because the probabilities associated to each of the possible
values of X have not been specified
Expected value = sum of all [x * P(x)]
P(x) = = 0.1
E(x) = (1 * 0.1) + (2 * 0.1) + (3 * 0.1) + (4 * 0.1) + (5 * 0.1) + (6 * 0.1) + (7 * 0.1) + (8 * 0.1) + (9 * 0.1) +
(10 * 0.1) = 5.5
Question 1
Suppose we randomly draw a single student from the population of psychology students at UM.
Jim and Peter are two psychology students at UM. Which of the following statements is correct?
A Jim and Peter are two elementary outcomes of this random experiment
B Jim and Peter are two independent events of this random experiment
C The probability of drawing either Jim or Peter is given by the product: P (Jim) * P (Peter)
For A: Since we draw only a single student (N = 1), every unique student counts as an elementary
outcome. If you draw multiple times, this would not hold true however! For instance if we draw two
students, an elementary outcome might be “Jim, Peter” or “Peter, Peter” or “Jim, Jim”.
For B: An event is a subset of the sample space, made up of one or more Elementary Outcomes.
Events are statistically independent when one event doesn’t influence the probability of the other
event taking place. The question doesn’t state if there were any events defined- or how, so we
cannot assess whether this is true.
For C: This is the formula “P(A and B) = P(A) * P(B)”. This formula however only holds true in case
the events are independent. As we just said for question B; we cannot assess whether Jim and Peter
count as independent events. Had answer B been true however, then would answer C have been
correct? No! They ask “the probability of drawing either Jim or Peter”. That would be P(A or B),
while the formula given refers to P(A and B). Had B been true, the formula for question C would
have to look like P(A or B) = P(A|B) + P(B|A) – P(A|B) * P(B|A).
(You can derive this long formula by looking at other findings we had in case of independent events,
such as P(A) = P(A|B), which you can then fill in to the default formula for P(A or B))
Question 2
Consider the population of UM students. Suppose that 40% of all students at the faculty of
economics are female. Additionally, 65% of all students at the faculty of psychology are female.
Suppose that we draw a simple random sample (SRS) of size n=250 students from the population
of UM students. We record 'specialization' and 'gender' of each student in the sample. Which of
the following quantities is a random variable for this random experiment?
A the sample size
B the percentage of economy students at the UM
C the number of women in the sample
For A: The sample size is not a random variable, it is already provided that our sample size N =
250.
For B: The economy students at the UM are a population, they aren’t the sample. The percentage
of economy students at the UM is thus not a random variable, but a known and fixed amount.
For C: In the sample we assess the specialization (economy/psychology) and gender of each
randomly drawn student, so the number of women within the sample are a random variable. Could
be that you draw 250 students, and 145 of them turn out to be women, but if you repeat the study,
next time you might find 200 of the students in your sample are women.
Question 3
A psychology faculty has 1000 students, of which 400 are men and the other 600 are women.
Furthermore, 500 of the 1000 students are Dutch, 300 are Belgian and 200 are German. We
randomly draw a single person from this population. This is a random experiment. Which of the
following statements is incorrect?
A ‘the age of the student drawn is equal to 20’ is an elementary outcome of this random
experiment
B ‘the age of the student drawn is equal to 20’ is an event of this random experiment
C ‘the age of the student drawn’ is a random variable
,For A: An elementary outcome is each unique combination that you could draw in your sample.
Among your 1000 students, it’s unlikely that only a single one of them happens to be 20 years old.
So you could have drawn perhaps 400 different students for who this statement would hold true.
For B: An event is a subset of the sample space, made up out of one or more Elementary
Outcomes. Focussing specifically on all students who have the specific age of 20, would definitely
count as an event.
For C: A random variable is made up out of one or more events. As we stated B to be true, “age is
equal to 20” is an event which is one of the possible values for your random variable.
Question 4
Suppose we draw a SRS of size n=250 from the population of UM students. We record
'specialization' and 'gender' of each student in the sample. Suppose that 40% of students at the
faculty of economics are female. Additionally, 65% of all students at the faculty of psychology are
female. What is the sample space of this random experiment?
A All 250 students that were drawn
B All possible samples of size n=250 that can be possibly drawn from the population of UM
students
C All psychology and economy students together
The sample space is the set of all possible elementary outcomes. (E.g. If you flip a coin twice, your
sample space would be (H,T); (H,H); (T,H); (T,T).)
The 250 students that were drawn are the random sample (in this case the simple random sample).
All psychology and economy students together would be your complete population.
Question 5
Refer to the population described in question 4. We now consider the random experiment in
which we draw a single student from this population. Which of the following probabilities can be
obtained based on the given information?
A the conditional probability of a male student, given that he studies psychology
B the unconditional probability of an economy student
C the conditional probability of an economy student, given that she is a female
For A: This works perfectly fine with the provided info. Nothing is said about males, but we
already know the probability P(Female|Psychology). Namely: If we know the student studies
psychology, there is a 65% chance that this person is female. We could then use the complement
rule to find this probability for males. So:
P(Male|Psychology) = 1 – P(Female|Psychology) = 35%.
((Do note that this only holds true since we only have two options here (male and female). For
example, if we used genders instead of biological sex, and considered that nonbinary genders are
acknowledged in this day and age, all psychology students would have been made up out of (man,
woman, non-binary). Then using the complement rule as we did above would simply show “the
probability of a student who is not a woman, given that they study psychology”.))
For B: For the unconditional probability, you would need to know the total amount of economy
students, as well as the total amount of students. Neither are given, so you cannot assess this.
For C: For this to be possible, the info given should have been the other way around, such as
“Suppose that 40% of the students who are female, study at the faculty of economics. From that
info, if you already know the student is female, you could infer the probability that she also studies
economics.
Question 6
Suppose we are conducting a random experiment and that two events A and B are not disjoint.
This implies
A P (A and B) = 0
, B P (A or B) = P(A) + P(B)
C A and B may be two independent events
For A: This only applies in case of disjoint events.
For B: This only applies in case of disjoint events.
For C: If an event is “not” disjoint but no further info is provided, the only thing we know, is that the
formulas applicable to disjoint events, do not apply. However this info is not sufficient to determine
whether our event is then independent or dependent, as both independent and dependent events
can be joint. (Disjoint events are dependent events, but of the extreme kind, so any other
dependent event is still joint.) So if for instance an answer option had been “P(A) = P(A|B)” this
would not have been a valid answer, as this one occurs only in case of independent events, and we
cannot infer whether our event is dependent or independent from the info in the question. So
answer C is the only legitimate option here; A and B may be two independent events, but you
cannot say this with certainty.
Question 7
Consider the following contingency table in which 1000 persons are classified according to their
age (young vs old) and marital status (unmarried, married, divorced)
We randomly draw a single person from this population. Which of the following statements is
correct?
A the events ‘the person is married’ and ‘the person is young’ are independent
B the probability of drawing someone who is old and divorced is 0.67
C the conditional probability of drawing a young person, given that he or she is divorced is 0.33
For A:
P(Young) = = 0.55
P(Married) = = 0.4
If the events are independent, P(A) = P(A|B) so P(Young) = P(Young|Married)
P(Young|Married) = = 0.5
So they are not independent!
For B:
P(old and divorced) = = 0.1
For C:
P(Young | Divorced) = = 0.33
Question 8
A box contains 10 cards numbered 1,2,...,10. From this box a single card is drawn randomly. The
random variable X is the number of the drawn card. The expected value of X is equal to
A 5.5
B6
C This value cannot be determined because the probabilities associated to each of the possible
values of X have not been specified
Expected value = sum of all [x * P(x)]
P(x) = = 0.1
E(x) = (1 * 0.1) + (2 * 0.1) + (3 * 0.1) + (4 * 0.1) + (5 * 0.1) + (6 * 0.1) + (7 * 0.1) + (8 * 0.1) + (9 * 0.1) +
(10 * 0.1) = 5.5
