Teaching Models

Probability

Samples, Surveys, and Bias
Most students have probably participated in some type of statistical survey, whether it was asking others for their opinions in order to make a decision, or participating in an on-line survey on the Internet. A survey asks people their opinions or experiences of a particular event or issue. All individuals fitting a particular description are called a population. A small group or part of a population is a sample. If that sample has characteristics similar to the entire population, it is said to be a representative sample.

A representative sample has no bias. A biased sample favors one particular answer. A convenience sample is taken of people that are easily available to complete the survey. A convenience sample is often biased because the sample population has some common attribute, such as all of them being teenagers or all being senior citizens. A random sample asks questions of all segments of the population. A random sample may or may not be biased, often depending upon how the question is asked.

If a representative sample is asked an unbiased question, then the results can be used to predict the response of the entire population. Proportions are used to make such predictions. Suppose 400 people out of 1,000 people surveyed answered yes to an unbiased sample question. Then a proportion can be used to predict how many people out of 60,000 would answer yes.

sample population v. total population

The prediction is that 24,000 people would answer yes to the survey.

Interpreting Statistics
When interpreting statistics, it is important to be alert for misleading statistics. Some examples of misleading statistics are given below.

Misleading Statements:
“Our survey shows 2 out of 3 investors in our company made money last year.”
Possible misleading factors:
The investors may have made money from some other investments, not from this company. The investors may have been salaried employees of the company.
Biased Survey Question:
“Should students be taught how to ride a skateboard to increase physical coordination?”
Possible misleading factor:
Developing physical coordination is a positive goal, thus it shows bias toward whatever is being suggested.
Biased Graphic Displays
skateboarding survey
Possible misleading factor:
This graph uses a scale that starts at 47%. It makes it look as if many more people answered no. When interpreting statistics, it is always important to read the data accurately as well as all of the scales and labels on a graphic display.

The Fundamental Counting Principle
The fundamental counting principle states that if an experiment or problem has two steps and there are m possible choices or outcomes for the first step and n possible choices or outcomes for the second step, then the total number of possible choices or outcomes for both steps is m × n. The set of all possible outcomes is called the sample space.

Probability
The result of an experiment is called an outcome. If the experiment is to toss a 1−6 number cube, then there are six possible outcomes, one for each face of the cube. An event is any collection of outcomes. Examples of events for tossing a number cube are that the number tossed is even, that the number is 1 or 2, or that the number is 3. The probability of an event is a measure of the likelihood that the event will occur. The probability is always a number between 0 and 1 that can be written as a fraction, a decimal, or a percent. A probability of 0 means that an event is impossible, while a probability of 1 means that an event is certain.

If each outcome is equally likely, the theoretical probability of an event is the ratio of the number of outcomes in the event to the total number of possible outcomes.

The experimental probability of an event is the ratio of the number of times the event occurs to the total number of experiments. This is an experimental estimate of the theoretical probability. For large numbers of experiments, the experimental probability approaches the theoretical probability of the event.

Computing Theoretical Probability
When a coin is tossed, there are two outcomes, heads or tails. Either outcome is equally likely. When a 1−6 number cube is tossed, each face is equally likely to turn up. When a marble is chosen from thoroughly mixed marbles of the same size, without looking, each marble has same chance of being chosen. Such outcomes are said to be equally likely outcomes.

Sometimes the word fair is used as in “fair coin” or “fair number cube.” To indicate that each outcome is equally likely, the word random is used, such as in saying “The object is chosen at random.”

When you toss two 1−6 number cubes, one way to record the outcome is to sum the numbers on the two cubes. The possible sums range from 2 to 12, but each sum is not equally likely. There are more ways to toss a sum of 7 than there are to toss a sum of 2. If the first cube shows a 3 and the second shows a 5, then this outcome can be represented by the ordered pair (3, 5). There are 36 such ordered pairs, each with an equally likely outcome. When all outcomes of an experiment are equally likely, the theoretical probability of event A is given by this ratio.

P(A) = number of outcomes in the event
total number of possible outcomes

Example:

Tossing a fair coin. List all outcomes: H (head), T (tail).
The probability of tossing a head is P(H) = one-half.

Example:

Tossing a fair 1−6 number cube. List all outcomes: 1, 2, 3, 4, 5, 6.
The probability of tossing an even number is the probability of tossing 2, 4, or 6.
P(2, 4, 6) = three-sixths = one-half

Example:

Tossing two fair number cubes. List all 36 equally likely outcomes.
two number cubes
The probability of tossing a sum of 7 is the probability of getting any of the outcomes
(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). So P(7) = three-thirty-sixths = one-sixth

Disjoint Events
If two events A and B have no outcome in common, then A and B are said to be disjoint events. When A and B are disjoint events with probabilities P(A) and P(B), respectively, the probability of A or B occurring is given by

P(A or B) = P(A) + P(B).

If the experiment is tossing two 1–6 number cubes, then the event S (getting a sum of 7) and the event L (getting a sum less than 7) are disjoint events. If event E is tossing an even sum, then S and E are disjoint events, but L and E are not disjoint since 2, 4, and 6 are even sums that are also less than 7.

For the events S, L, and E above,

P(S) = one-sixth, P(L) = five-twelfths , and P(E) = one-half
P(S or L) = P(S) + P(L) = seven-twelfths
P(S or E) = P(S) + P(E) = two-thirds

The Probability of not A
If A is an event, then the event not A consists of all outcomes that are not in A. The events A and not A are disjoint events. The events A and not A include all possible outcomes, so the probability is 1. Thus P(not A) = 1 − P(A).

Example:

What is the probability of tossing two 1−6 number cubes and not tossing a sum of 7?
1 − P(7) = 1 − one-sixth = five-sixths

Dependent and Independent Events
A compound event consists of the outcomes of two or more events. In the compound event A and B, A and B are independent events if the outcome A has no influence on the outcome of B. If the outcome of A does have an influence on the outcome of B, then A and B are dependent events.

If A and B are independent events, then the probability of A and B occurring is simply the product of the probability of A and the probability of B.
P(A and B) = P(A) × P(B)

Example:

Toss a fair coin twice. What is the probability of getting two heads in a row? Since the outcome of the first toss has no influence on the outcome of the second toss, the probability is simply P(H and H) = one-half × one-half = one-fourth

If A and B are dependent events, then the probability of A and B is the product P(A) × P(B given that A has occurred).
P(A and B) = P(A) × P (B given that A has occurred)

Example:

A jar contains 5 red and 10 white marbles. Find the probability of drawing, at random, two red marbles in a row if the first marble is not returned.

Event A: The first marble drawn is red.
Event B: The second marble drawn is red.

P(A) = five-fifteenths = one-third

Now find P(B) assuming that the first marble chosen is red.

P(B given that A has occurred) = four-fourteenths = two-sevenths
P(A and B) = P(A) × P(B given that A has occurred) = one-third × two-sevenths = two over twenty-one.


Teaching Model 19.2: Theoretical Probability


Houghton Mifflin Math Grade 6