EDUC221
A Few Notes About Probability and Statistics
–Some basics. Statistics is the science or study of data. Statistical studies require collecting,
sorting, representing, analyzing, and interpreting information. The information is then used for
predicting, drawing inferences, and making decisions. There is always some degree of
uncertainty with data that is collected. This means that statistics is most often concerned with
using information in the face of uncertainty. Probability gives a way to measure uncertainty
and is therefore essential to understanding statistical methods.
–Why Teach Probability and Statistics?
*We are flooded with data that demands an increasing understanding of statistics for us
to understand its use. For example: Which money market or retirement fund makes the most
sense? What does a particular advertising claim really mean? What effect will the rate of
population growth have on our lives? . . .
*Learning probability and statistics provides real applications of arithmetic. When basic
computational skills are used in a context, students have the opportunity to see the advantages
and limitations of their calculations.
*Studying probability and statistics also helps students develop critical thinking skills. In
carrying out experiments in probability and statistics, students develop ways to cope with
uncertainty as they search for the truth in a situation and learn to report it faithfully.
Approaching situations statistically can help students face up to prejudices, think more
consistently about arguments, and justify their thinking with numerical information. This
approach has applications in all areas of life–social, political, and scientific, . . .
-What concepts in probability and statistics should be taught?
The teaching of probability and statistics should stem from real problems. A theoretical
or abstract approach is not appropriate for the elementary grades. The approach should be
based on experiments that draw on children’s experience and interests. Children’s intuition
needs to be challenged first. Once they sense what “should happen:” in a situation, then it’s
timely for them to carry out an experiment to test their predications. Not only will such
experiments provide firsthand experience in collecting, organizing, and interpreting data, but
they will also reinforce computational skills. Basic to probability and statistics are the following
ideas:
1. Collecting Data.
Students need experience gathering both factual data and data that involves
opinions. The latter requires that students consider how to collect data in a way that avoids
biased responses.
2. Sampling.
In real life, even though information is often needed about an entire population,
it may only be possible to sample a part of the population. That information from the sample is
then used to infer characteristics about the total population. Students need to learn the
difference between random and nonrandom samples and the importance this difference makes
in statistical studies. For example, to determine the percentage of people in the general
population who are left-handed, you would not poll professional baseball players as a random
sample. (Why?)
3. Organizing and Representing Data.
Students need experience organizing data and representing it graphically in a
variety of graphs, tables, and charts.
4. Interpreting Data.
Students should learn to read graphs, making quick visual summaries as well as
further interpretations and comparisons of data through finding means, medians, and modes.
5. Assigning Probabilities.
Initial experiences with measuring uncertainty should be informal and should
include discussion of whether a result is possible or likely, or whether outcomes are equally or
not equally likely. Assigning probabilities gives further information for making a decision in the
face of uncertainty. The probability of an event can be represented by a number from 0 to 1.
For example, the probability of rolling a 4 when rolling a dies is 1/6. The probability of rolling
an even number when rolling a die is 3/6. The numerator of the fraction represents the
number of outcomes you’re interested in; the denominator represents the total number of
possible outcomes.
6. Making Inferences.
Students need to draw conclusions based on their interpretation of data. They
should learn to justify their thinking using numerical information they’ve collected and
analyzed.
Probability
1.
What is probability?
Probability is really a measurement, a measurement of how likely a particular event is to
happen or the chance that the event will take place. A useful picture of probability is:
impossible
less likely
equally likely
0%
←
50%
0
more likely
→
½
0
certain
100%
1
0.5
1
Probabilities are represented as a number between 0 and 1 in either decimal, fraction, or
percent form.
2.
Determining probabilities.
a. Experimental or empirical probability (observed): This kind of probability is
determined by observing outcomes of experiments. These probabilities cannot be
accurately predicted with only a small number of trials.
Note: How are experimental probabilities helpful? Sometimes it’s hard to determine the actual
probability of a complicated event and so we do a simulation instead. That is, we do the
experiment a large number of times and estimate the probability we want by the fraction of the
experiments in which the event occurred.
b. Theoretical probability (abstract): This kind of probability is what is likely to happen
under ideal conditions, over the long run.
Determining a theoretical probability for an event A:
For an experiment with a set of equally likely outcomes, the probability of an event A is given
by:
P(A) =
(Number of ways event A could happen)
= n(A)
(Number of possible outcomes of the experiment)
n(S)
where A is the set of desired outcomes and S is the set of all possible outcomes.
3.
Definitions:
probabilities: are ratios, expressed as fractions, decimals, or percents that measure
the likelihood that an event is to occur–determined by considering the results or
outcomes of experiments (either empirically or theoretically)
experiment: an activity under consideration, such as tossing a coin or rolling a die
or drawing a card...
outcome: a possible result of an experiment
sample space: the set of all possible outcomes for an experiment
event: subset of a sample space
equally likely: when one outcome is just as likely to occur as another
chance:
Example experiments:
Experiment–Tossing a fair coin and observing whether it lands on H or T.
Sample space: {H, T}
Event: getting a tail, {T}
P(T)= n(T) = 1
n(S) 2
Experiment–Rolling a fair number cube and observing whether it lands on an even
number or an odd number.
Sample space:
Event: rolling an even number
P(E)=
4.
Let’s try finding probabilities for an experiment:
–Experiment: Pick one card at random from a standard deck of 52 cards. (Each card has an
equal chance of being drawn.)
Find the following probabilities:
a) The event A that a heart is drawn.
b) The event B that a card less than 2 is drawn.
c) The event C that a card less than 2 and greater than 9 is drawn.
d) The event D that a heart or a club is drawn.
e) The event E that a diamond and a queen is drawn.
For our Reflection
1. Take a look at SUM GAME through the lens of the process standards: Mathematics as
Problem Solving; Mathematics as Communication; Mathematics as Reasoning; Mathematics as
Representation; and Mathematical Connections . . . . What did you observe about the task of
analyzing SUM GAME respective to each of these standards? (Refer to your own experiences,
observations, discussions, thoughts, questions, ....) Include specific examples.
2. Take a look at SUM GAME mathematically. Is the game fair? Thoroughly justify your
response. Now, take a look at a new game–PRODUCT GAME–played the same way except by
taking the products of the numbers on the dice instead of the sum. Is this a fair game? Why or
why not? Provide a thorough justification of your response.
3. Create a content/concept map that illustrates topics relevant to the SUM GAME activities,
discussions, and possible extensions that you might consider. Include mathematical concepts,
connections to other mathematics, connections to other content domains, and teaching issues.
Include examples for each of the areas of your map.
4. A student claims that if a fair coin is tossed and comes up heads five times in a row, then the
probability of tails on the next toss is greater than the probability of heads. What is your reply?
Consider the underlying mathematical issues as well as how you might handle the situation
pedagogically.
5. Experiment: A cookie jar contains 2 raisin cookies, 4 oatmeal cookies and five chocolate chip
cookies. You will reach in and draw one cookie at random–so each cookie is equally likely to be
drawn.
*What is the sample space of the experiment?
*If a cookie is drawn at random, what is the probability it will be chocolate chip?
*What is the probability that a cookie drawn at random will be either oatmeal or raisin?
*What is the probability of not getting a chocolate chip cookie when one is selected at
random?
6. a. Using a tree diagram, determine the total number of possible ways that a family can
consist of four children (if order does matter).
b. If order does not matter, how many boy/girl combinations are possible with 4 children?
(Use your findings from part “a” to help.)
c. What is the probability that a family with four children has 3 boys and 1 girl?
7. How many phone numbers are possible with a 649 prefix?