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Basic Ideas in Probability Study Guide (page 2)

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Updated on Oct 5, 2011

Example

A highway patrolman randomly selects a car on the interstate and measures its speed by radar. The patrolman may decide to let the car continue, issue a warning, or issue a ticket.

  1. Give the sample space.
  2. List all events.
  3. The patrolman issued a ticket.Which event(s) has (have) occurred?
  4. Are the outcomes for this random experiment equally likely?

Solution

  1. S = {C,W, T} where C represents the outcome of the car continuing, W is the outcome of the patrolman issuing a warning, and T is the outcome of the patrolman issuing a ticket.
  2. Because there are three possible outcomes, there are 23 events: ø, {C}, {W}, {T}, {C,W}, {C,T}, {W,T}, S.
  3. {T}, {C,T}, {W,T}, S.
  4. It is very unlikely that all three outcomes are equally likely. For example, if a patrolman stops a car, it is generally more likely that a ticket and not a warning will be issued.

Definition of Probability

Now that sample space and events have been defined for a random experiment, we need to consider how to associate probabilities with all possible events. The classical definition of probability is as follows: The probability of an event E, denoted by P(E), is the ratio of the number of outcomes favorable to E to the total number of outcomes in the sample space S. For our die example, suppose we are interested in the probability of the event A, rolling an even number. The number of outcomes favorable to A is three because the outcomes "2,""4," and "6" are all possible outcomes that are even. The total number of outcomes in the sample space is six. Thus, by the classical definition of probability,

.

It is important to note that this method for calculating probabilities is appropriate only when the outcomes of an experiment are equally likely. Further, both the event A and the sample space S must have a finite number of outcomes.

Sample spaces for many experiments can be constructed so that the outcomes are equally likely. However, this is not always possible. Suppose that an airline wants to know what proportion of the calls it receives on its reservation line results in a reservation. The sample space consists of only two outcomes: A reservation is made and a reservation is not made. It is very unlikely that these outcomes are equally likely. To determine what the probability is that a randomly selected call will result in a reservation, the company could record calls and determine whether or not each call resulted in a reservation. Then, based on a large number of calls, the proportion resulting in reservations would provide an estimate of the probability of interest. This is an example of the relative frequency definition of probability.More formally, for the relative frequency definition of probability, the probability of an event E, denoted by P(E), is defined to be the relative frequency of occurrence of E in a very large number of trials of a random experiment; that is, if the number of trials is quite large:

 

Applying the Classical and Relative Frequency Definitions

Classical Definition of Probability

The probability of an event E, denoted by P(E), is the ratio of the number of outcomes favorable to E to the total number of outcomes in the sample space S.

Relative Frequency Definition of Probability

The probability of an event E, denoted by P(E), is defined to be the relative frequency of occurrence of E in a very large number of trials of a random experiment.

It is important that there be a very large number of trials for the relative frequency to accurately reflect the probability of interest. The reason for this is that the relative frequency can fluctuate quite a bit for a small number of trials. For example, suppose that the probability that a randomly selected call to the airline results in a reservation is 0.4. Suppose we use the relative frequency definition of probability to find this probability. In Figure 9.1, we graph the relative frequency for a particular set of trials as the number of trials increases up to 100. Notice how much fluctuation occurs with only a few trials, but that the relative frequency tends to stabilize around 0.4 as the number of trials increases.

Figure 9.1

Looking both at the classical and relative frequency definitions of probability,we can observe some characteristics that are true for all probabilities.

  • Axiom 1: Every probability must be between 0 and 1; that is, for any probability p, 0 ≤ p ≤ 1.
  • Axiom 2: Because the sample space is the set of all possible outcomes, one of the outcomes in S, and thus, the event S, must occur any time the experiment is conducted. Therefore, the probability of the sample space is one (i.e., P(S) = 1).
  • Axiom 3: If two events E and F are disjointed (have no outcomes in common), then P(E or F) = P(E) + P(F) because, using the classical definition of probability,

The third characteristic also holds if there are more than two disjointed sets.

These three characteristics are called the axioms, or rules, of probability. Using the three axioms of probability, more results of probability can be obtained. Three of these follow.

  • Result 1: The null event has no outcomes in the sample space and thus has probability zero of occurring (i.e., P(φ) = 0).
  • Result 2: For any event E, the complement of E is the set of all outcomes in the sample space that are not in E. The probability of the complement of E is 1 – P(E). To see this, note that P(E) + P(not E) = 1, because every outcome in the sample space is in E or the complement of E. By subtraction, we then have P(E) = 1 – P(not E).
  • Result 3: For any events E and F, P(E or F) = P(E) + P(F) – P(E and F). To see this, notice that if we add the probability of E to the probability of F as we did for axiom 3, we have added the probability associated with the outcomes that are in both events twice, once for E and once for F. Thus, we must subtract out the probability of the outcomes the two events have in common to correctly compute the probability of E or F.

Using these basic ideas, we can find probabilities for a number of experiments.

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