Probability A short story

Sample Spaces

As we saw in the previous section, probability questions arise when we are faced with a situation that involves uncertainty. Such a situation is called a random experiment, an experiment that produces an outcome that cannot be predicted in advance (hence the uncertainty).

Here are a few examples of random experiments:

  1. Toss a coin once and record whether you get heads (H) or tails (T). The possible outcomes that this random experiment can produce are: {H, T}.
  2. Toss a coin twice. The possible outcomes that this random experiment can produce are: {HH, HT, TH, TT}.
  3. Toss a coin 3 times. The possible outcomes in this case are: {HHH, THH, HTH, HHT, HTT, THT, TTH, TTT}.
  4. Toss a coin until you get the first tails (T). When we conduct this experiment, one possible outcome is that we get T in the first toss and we are done. Another possible outcome is that we get H in the first toss, toss a second time, get T and be done. We might need three tosses until we get the first T, etc. The possible outcomes of this random experiment are therefore: {T, HT, HHT, HHHT, …}. (Note that in this example the list of possible outcomes is not finite as in examples 1-3. This is not an important distinction at this point, just a noteworthy observation.)
  5. Choose a person at random and check his or her blood type. In this random experiment the possible outcomes are the four blood types: {A, B, AB, O}.
  6. There are two job openings for a staff position at a certain college, and 4 equally qualified candidates for the job (Ann, Beth, Jim and Dan). For fairness, the human resources department decides to choose two of the four candidates at random. The possible outcomes of this random experiment are all possible pairs of candidates: { (Ann, Beth), (Ann, Jim), (Ann, Dan), (Beth, Jim), (Beth, Dan), (Jim, Dan) }.

Comment: Does Order Matter?

Note that when a coin is tossed twice, as in example 2, the possible outcome HT (indicating that the first toss was H and the second T) is NOT the same as the outcome TH (indicating that T occurred first and then H), and therefore both outcomes were listed separately. This is an example of a situation when order does matter. However, order does not always matter. Example 6 is a case in which order does not matter. The outcome (Ann, Beth) indicates that Ann and Beth are the two randomly chosen to get the jobs. Whether Ann appears first or Beth does is irrelevant in this case, and therefore (Beth, Ann) was not listed as a separate outcome.

There is really no rule that dictates when order matters and when it doesn’t. It is sometimes clear from the way the random experiment is defined. For example, suppose I were to change example 6 slightly:

There are two job openings for similar staff positions at a certain college: one in the Registrar’s Office, and one in the Office of Admissions. The Human Resources Department has identified four equally qualified candidates for the jobs (Ann, Beth, Jim and Dan), and for fairness decides to choose two of the four candidates at random. The first chosen will fill the position in the Registrar’s Office, and the second will fill the position in the Office of Admissions.

Now order is relevant—the two outcomes (Ann, Beth) and (Beth, Ann) are not the same in this scenario. The first outcome indicates that Ann got the position at the Registrar’s Office and Beth got the position at the Office of Admissions, while the second outcome indicates the reverse. In this case, therefore, all the possible outcomes are:

{ (Ann, Beth), (Beth, Ann), (Ann, Jim), (Jim, Ann), (Ann, Dan), (Dan, Ann),

(Beth, Jim), (Jim, Beth), (Beth, Dan), (Dan, Beth), (Jim, Dan), (Dan, Jim) }

Each random experiment has a set of possible outcomes, and there is uncertainty as to which of the outcomes we are actually going to get once the experiment is conducted. This list of possible outcomes is called the sample space of the random experiment, and is denoted by the (capital) letter S.

Going back to the 6 examples above, we can write:

Example 1: S = {H, T}

Example 2: S = {HH, HT, TH, TT}

Example 3: S = {HHH, THH, HTH, HHT, HTT, THT, TTH, TTT}

Example 4: S = {T, HT, HHT, HHHT, …}

Example 5: S = {A, B, AB, O}

Example 6: S = { (Ann, Beth), (Ann, Jim), (Ann, Dan), (Beth, Jim), (Beth, Dan), (Jim, Dan) }.

Events of Interest

So far, we have a random experiment and its sample space—the set of all possible outcomes it can produce. Where does probability come into the picture?

Once we have defined a random experiment, we can talk about an event of interest, which is a statement about the nature of the outcome that we’re actually going to get once the experiment is conducted. Events are denoted by capital letters (other than S, which is reserved for the sample space).

Example: Tossing a Coin 3 Times

Consider example 3, tossing a coin three times. Recall that the sample space in this case is:

S = {HHH, THH, HTH, HHT, HTT, THT, TTH, TTT}

We can define the following events:

Event A: “Getting no H”

Event B: “Getting exactly one H”

Event C: “Getting at least one H”

Note that each event is indeed a statement about the outcome that the experiment is going to produce.

In practice, each event corresponds to some collection (subset) of the outcomes in the sample space:

Event A: “Getting no H” → TTT

Event B: “Getting exactly one H” → HTT, THT, TTH

Event C: “Getting at least one H” → HTT, THT, TTH, THH, HTH, HHT, HHH

Here is a visual representation of events A, B and C.

We have a large rectangle labeled "S" which represents the entirety of the sample space. Inside this rectangle we have two circles labeled "A" and "C." A contains only "TTT". Inside of C, we see "HHH," "THH," "HTH," "HHT," and a circle representing event B. Inside B are "HHT," "THT," and "TTH." Note that all of the items inside B are also inside C, so C fully encloses B.

From this visual representation of the events, it is easy to see that event B is totally included in event C, in the sense that every outcome in event B is also an outcome in event C. Also, note that event A stands apart from events B and C, in the sense that they have no outcome in common, or no overlap. At this point these are only noteworthy observations, but as you’ll discover later, they are very important ones.

Once an event is defined, we can talk about the probability that it will occur. So, if we have defined an Event A, we can use the notation we previously mentioned to represent its probability, namely P(A).

The following figure summarizes the information in this section:

A flow chart. We start with a Random Experiment, which leads to a Sample Space, called S. Then, we define Events (A, B, C...) which correspond to a collection of outcomes in S. After that, we obtain the probability of the events, resulting in P(A), P(B),...

Equally Likely Outcomes

In the Probability: Introduction section we learned how the relative frequency approach can be used to estimate the probability of an event. While sometimes this is the only method that can be used to estimate probability (such as when figuring out the probabilities of the occurrence of different blood types among the population), this method requires a lot of time and effort, especially since in order to get reliable estimates we need to repeat the random experiment many times. We are now moving on to a different method, which can be applied in cases in which the random experiment produces outcomes that are all equally likely. We’ll start with a simple example to introduce the idea of the method, and then move on to more interesting examples.

Example: Rolling a Fair Die

When an ordinary fair die is rolled once, what is the probability that the number rolled is even? We’ll denote this event by E (for even), so we are interested in finding P(E). Let’s analyze this problem:

* The random experiment is rolling a fair die once.

* The sample space of all possible outcomes in this case this is S = {1, 2, 3, 4, 5, 6}.

* Since the die is fair, this means that all 6 possible outcomes are equally likely (each having a probability of 1/6 of occurring)

* We are interested in a particular type of outcome, which is represented by event E—getting an even number.

Since 3 out of the 6 equally likely outcomes make up the event E (the outcomes {2, 4, 6}),

The 6 possible outcomes of rolling a die. These are 1, 2, 3, 4, 5, or 6. Of these, 2, 4, 6 are in our event E.

the probability of event E is simply P(E) = 3/6.


Let’s Generalize

In the special situation where all the outcomes in S are equally likely, we can find the probability of any event A by dividing the number of outcomes in A by the number of outcomes in S:

P(A) = (number of outcomes in A)/(number of outcomes in S)

The purpose of the next activity is to give you guided practice on how to find the probability of an event in situations in which all the possible outcomes are equally likely.

Scenario: Gender of Children

A couple is planning to have 3 children. Assuming that having a boy and having a girl are equally likely, and that the gender of one child has no influence on (or, is independent of) the gender of another, what is the probability that the couple will have exactly 2 girls?

The “random experiment” in this case is having 3 children, as odd as that may sound in this context. The next and most important step is to determine what all of the possible outcomes are, and list them (i.e., list the sample space S). In this case, each outcome represents a possible combination of genders of 3 children (note that examples with the same number of boys and girls but a different birth order must be listed separately).

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