Robert Bartoszynski - Probability and Statistical Inference

(2.6)

Proof : We will proceed by induction. The theorem is true for . Assume now that formula ( 2.6) holds, and write

Since

we have

Applying formula ( 2.6) to and to , and then combining the corresponding terms, we obtain ( 2.6) with replaced by .

1 2.4.1 Let events , , and be such that contains and is disjoint from . Moreover, is twice as likely as , three times as likely as , and half as likely as its complement . Find .

2 2.4.2 Events , , and are such that while is not empty. Determine as a function of if .

3 2.4.3 Find for events and such that , and .

4 2.4.4 To make the formula valid, must equal: (1) . (2) . (3) . (4) None of the above.

5 2.4.5 Three events , , and are such that is contained in the intersection , and . Find: (i) . (ii) The probability that exactly two of the events , , will occur. (iii) The probability that exactly one of the events will occur.

6 2.4.6 Let and be five events such that and . Moreover, at least one of the events must occur, and the intersection of any three events among is empty. (i) Find if . (ii) Omit the assumption that at least one of the events must occur and determine all possible values of if it is known that .

7 2.4.7 If events are such that , while every triple intersection is empty, find the smallest possible value of the probability of intersection .

8 2.4.8 Before the era of cell phones (a time that most of today's students do not remember), people often used public phones that operated after a coin was inserted. These phones were sometimes faulty and either took a coin without giving a connection or gave the connection but also returned the coin. (i) A faulty public phone returns the coin with probability 60%, it gives you the number you dial with probability 20%, and it takes your coin and does not give you the required connection with probability 30%. Find the probability that you will talk with the number you dial for free. (ii) A certain public phone is such that it returns the coin with probability , connects you with the number you dial with probability , and it gives you the connection for free with probability . Let us agree to say that the phone is individually honest if it takes your money if and only if it provides you with the required connection, and that it is socially honest if it takes, on average, as many coins as it gives correct connections (but perhaps from different customers). Find conditions for , and under which the phone is individually honest. (iii) Find conditions under which the phone is socially honest.

9 2.4.9 Based on a survey of undergraduate sophomores at a certain university, the researchers found out that 67% of students live in a dorm (), 52% are interested in studying abroad (), and 48% are planning to attend graduate school (). Furthermore, 32% live in a dorm and are planning studying abroad, 30% are planning to study abroad and continue their education in a graduate school, while 30% live in a dorm and plan to apply to graduate schools. Moreover, 20% of sophomores answered all three questions positively (). What is the probability that a randomly selected sophomore at this university either lives in a dorm, or is thinking of studying abroad, or is planning to apply to a graduate school?

10 2.4.10 A die is loaded in such a way that the probability of dots on the top face is proportional to , for . What is the probability that in one roll of the die an odd number of dots will turn up?

For the so‐called classical or logical interpretation of probability, we will assume that the sample space contains a finite number of outcomes and all of these outcomes are equally probable .

Obviously, in this case, each of the outcomes has the same probability and for every event

(2.7)

In many real situations, the outcomes in the sample space reveal a certain symmetry, derived from physical laws, from logical considerations, or simply from the sampling scheme used. In such cases, one can often assume that the outcomes are equiprobable and use ( 2.7) as a rule for computing probabilities. Obviously, the function in ( 2.7) satisfies the axioms of probability.

To use some very simple examples, in tossing a regular die, each face has the same probability . Then the probability of the event is , since there are three odd outcomes among the possible six.

The case above is rather trivial, but considerations of symmetry can sometimes lead to unexpectedly simple solutions of various problems.

Peter tosses a fair coin times, and Paul tosses it times. What is the probability that Paul tosses more heads than Peter?