Next chapter: Normal distribution

For some purposes, probability is best thought of as subjective. Questions such as "What is the probability that Boston will defeat New York in an upcoming baseball game?" cannot be calculated by dividing the number of favorable outcomes by the number of possible outcomes. Rather, assigning probability 0.6 (say) to this event seems to reflect the speaker's personal opinion --- perhaps his or her willingness to bet according to certain odds. Such an approach to probability, however, seems to lose the objective content of the idea of chance; probability becomes mere opinion. Two people might attach different probabilities to the outcome, yet there would be no criterion for calling one "right" and the other "wrong." We cannot call one of the two people right simply because he or she assigned a higher probability to the outcome that actually occurred. After all, you would be right to attribute probability 1/6 to throwing a six with a fair die, and your friend who attributes 2/3 to this event would be wrong. And you are still right (and your friend is still wrong) even if the die ends up showing a six!

The following example illustrates the present approach to probabilities. Suppose you wish to know what the weather will be like next Saturday because you are planning a picnic. You turn on your radio, and the weather person says, “There is a 10% chance of rain.” You decide to have the picnic outdoors and, lo and behold, it rains. You are furious with the weather person. But was he or she wrong? No, they did not say it would not rain, only that rain was unlikely. The weather person would have been flatly wrong only if they said that the probability is 0 and it subsequently rained. However, if you kept track of the weather predictions over a long periods of time and found that it rained on 50% of the days that the weather person said the probability was 0.10, you could say his or her probability assessments are wrong.

So when is it sensible to say that the probability of rain is 0.10? According to a frequency interpretation, it means that it will rain 10% of the days on which rain is forecast with this probability.

Next chapter: Normal distribution