| Scatter plots, tables, graphs indicate "trends", if they exist. When a trend is identified, a future prediction can be made with a certain level of accuracy. Prediction of future events is a primary use for math models. The level of accuracy depends on the amount of correlation between the data points. A "perfect" correlation would have a 100% predictabiility value and would appear as a perfect line (or other functional relationship). Mathematicians do not use "magic" or "crystal balls" to predict the future - they use "facts" combined with an inductive approach which assumes that things will continue along the same path in the future. So, now we begin our study of Probability (or predicting future events). Probability is defined as the likelihood of an event occurring. This likelihood may be expressed as a fraction, decimal, or percent. In order to calculate the probability using formulas you must know the total number of outcomes (called Sample Space). The sample space is an actual list of outcomes which may be counted. You should have some experience with simple probability, using spinners, dice, etc. You should also have some exposure to geometric probability which compares areas of target regions. A quick review of these topics is included here...
There are two major categories for calculating probability: Theoretical Probability - uses numbers and formulas Experimental Probaility - based on collected data from experiments or "simulations" In plain English, the formula for Theoretical Probability is to divide the number of desired outcomes by the total number of outcomes in the Sample Space (where both numbers are provided to you). In the language of Math,
Experimental Probability is calculated using the same formula BUT the data is gathered differently (data is recorded from repeated physical trials). The greater the number of trials, the more reliable the prediction will be. Patience is required to perform enough trials to be truly useful. Enter a wonderful use for computers and programs...Simulations. Simulations are used when a very complex event must be modeled. Since probability is based on random events, the data must be gathered to insure preservation of random occurrences. The random number generator function of a graphing utility or a random number table may be used to represent data. A simulation is designed using the notion of "yes/no" assignment of values and then counting to get values. Below is a table of values representing 10 families containing 4 children each where even numbers represent girl children and odd numbers represent boy children.  We can use this simulated data to answer many questions, like what is the probabiility that a 4 child family will contain exactly 2 girls. After counting the number of groups containing exactly 2 even numbers, we get 4 groups. However, 10 total groups were represented. So.... the probability of a 4 child family containing exactly 2 girls is 4 out of 10 or , in reduced form, 2 out of 5 (2/5 = .4 = 40%) Note: Other questions could involve analyzing at least, at most, or all same sex children. Greater accuracy could be achieved by using more than 10 random number groupings. |
