Chapter II Part I - Statistics I - Exercises

This is a set of exercises created by the teaching Faculty for Statistics I, Chapter II - Random Variables, from the Lisbon Accounting and Business School.

Questions

Question 1

Consider the following events:

1. “get a 7 when rolling a cubic die”
2. “Spain win the next World Cup”
3. “rain in London”

The correct choice is: A is impossible, while C is unlikelyB is certain, and C is very likely\(P(B)\) and \(P(C)\) are computed using the subjective notion of probabilityA and C are impossible

Question 2

When computing a probability from the analysis of all possible outcomes (equally likely) it was used: the frequentist definition of probabilitythe classic definition of probabilitythe subjective definition of probabilitythe axiomatic definition of probability

Question 3

Let \(\Omega\) be the outcome space associated to a random experiment. Let \(A\) and \(B\) two events (\(A\subset\Omega\) and \(B\subset \Omega\)) such that \(0<P(A)<1\) and \(0<P(B)<1\). You know that \(A\subset B\). What is \(P[(A\cup B)\cap B^c]\)?

0P(A)P(B)1

Question 4

Let \(A,B,C\) events defined in \(\Omega\) such that:

1. \(A\cup B\cup C=\Omega\)
2. \(P(A)=0.3\)
3. \(P(C)=0.5\)
4. \(P(B^c)=0.7\)
5. \(A\cap B=\emptyset\)
6. \(B\cap C=\emptyset\)

TRUEFALSE \(P(A\cap C)=0.1\)

Question 5

Let \(A\) and \(B\) two independent events defined in the same \(\Omega\), where \(A\) has a probability twice as large as \(B\). Knowing that \(0.5\) is the probability that at least \(A\) or \(B\) happens, you have: a) \(P(B)=\frac{3+\sqrt{5}}{4}\) b) \(P(B)=\frac{3-\sqrt{5}}{4}\) c) \(P(B)=\frac{3+\sqrt{5}}{4}\) or \(P(B)=\frac{3-\sqrt{5}}{4}\) d) None of the above

abcd

Question 6

Let \(A\) and \(B\) be two events in the same outcomes space \(\Omega\) such that \(P(A)=0.75\), \(P(B)=0.5\) and \(P(A\cup B)=1\). Then, \(P(A|B)\) is equal to:

13341214

Question 7

Let \(A\) and \(B\) two events in the same sample space \(\Omega\). You know that \(P(A)=a\) and \(P(B)=b\), with \(0<a<1\) and \(0<b<1\). The probability that neither \(A\) nor \(B\) happen is:

1. TRUEFALSE \((1-a-b+ab)\) if \(A\) and \(B\) are independent events.
2. TRUEFALSE \(1\) if \(A\) and \(B\) are complementary events.

Question 8

A financial institution offers two assets to invest (\(A\) and \(B\)) with very attractive returns. It is known that from the pool of costumers, 10% invests a share of their capital in asset \(A\), and the rest invests in \(B\). From those who invest in \(A\), 70% get above the market returns. From those who do not, only 55% get returns above the market. If you select, randomly, a client of this firm:

1. TRUEFALSE The probability of getting a return above the market is 0.565.
2. TRUEFALSE Knowing that a client got an above the market return, the probability this client invested in \(B\) is equal to 0.495.
3. TRUEFALSE The events “investing in asset A” and “getting a return below the market” are independent.

Question 9

Assume that 5% of students will be excluded of continuous assessment, and from these, 98% will have final grade above 13. From those not excluded from the continuous assessment, 10% will have a final grade below 13. If you select randomly a student at the end of the semester:

1. The probability of that student having a final grade above 13, given it was in continuous assessment is: 0.900.1440.936None of the above
2. Knowing that a student got a final grade above 13, the probability that the student was excluded from the continuous assessment is equal to: 0.020.05420.1None of the above

Question 10

In the class for Management from some Higher Education institute, 70% of the students are male. From the male students, for 60% was their first choice. From the females, 75% chose this course as their first choice. Selecting randomly a student in this Management course, let \(p\) be probability of being a student that chose this course as her/his first option.

1. The value for \(p\) is: 0.2250.420.645None of the above
2. The probability of being a female, knowing that for this student this course was her first choice. 0.75/p0.225/p0.42/pNone of the above
3. The probability of a student being a male, or that the student selected management as their first choice is 0.7p0.28+p0.7+pNone of the above
4. TRUEFALSE The events “being a male” and “not having selected management as her/his first choice” are independent.