Chapter 6
Section 1
The Study of Randomness
How often would this method give a correct
answer if I used it very many times?
• If we know the blood types of a man and a
woman, what can we say about the blood
types of their future children?
• Give a test for AIDS to the employees of a
small company. What is the chance of at
least one positive test if all the people
tested are AIDS free?
The mathematics of probability
begins with the observed fact that
some phenomena are random – that
is, the relative frequencies of their
outcomes seem to settle down to
fixed values in the long run.
Graph on pg. 313
Random does not mean haphazard
– it describes a kind of order that
emerges only in the long run.
-- if individual outcomes are un
certain but there is nonetheless a
regular distribution of outcomes in a
large number of repetitions.
Probability
The probability of any outcome of a random
phenomenon is the proportion of times the
outcome would occur in a very long series
of repetitions.
Mathematical probability is an idealization
based on imagining what would happen in
an indefinitely long series of trials.
Independence
The outcome of one trial does not influence
the outcome of any other trial.
Random Phenomenon
• has outcomes that we cannot predict but
that nonetheless have a regular
distribution in very many repetitions.
Imagine a spinner with three sectors, all the same size, marked 1, 2, and
3 as shown on page 311.
Imagine spinning this spinner 3 times and recording the numbers as
they occur.
You want to determine the proportion of times that at least one digit
occurs in its correct position.
i.e. 1 2 3 – all digits in correct position
1 3 2 – one digit in its correct position
3 1 2 – none in correct position
Use your calculator to randomly generate the three digits.
Randint(1,3,3)
Continue to press enter to generate more three digit combos.
Use a tally mark to record the results in a table like below.
Do 20 trials and then calculate the relative frequency for the event “at least
one digit in the correct position.”
At least one digit
In the correct position____________________________________
Not
Combine your results with everyone in class to obtain as many trials as possible.
Put SPIN123 into your calculator from page 312.
Run the program with 25 trials, 50 trials, and then 100 trials.
The actual probability that at least one digit will occur in its correct place is
19/27 = .7037 or approx. 70%.
How does that compare to your data?
70