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 – all digits in correct position – one digit in its correct position – 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?

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