Probability Explanation And Exercises Worksheet Page 2
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37Remarks on the Concept of “Probability”
by Dan Osherson
Prerequisites
• None
Learning Objectives
1. Define symmetrical outcomes
2. Distinguish between frequentist and subjective approaches
3. Determine whether the frequentist or subjective approach is better suited for a
given situation
Inferential statistics is built on the foundation of probability theory, and has been
remarkably successful in guiding opinion about the conclusions to be drawn from
data. Yet (paradoxically) the very idea of probability has been plagued by
controversy from the beginning of the subject to the present day. In this section we
provide a glimpse of the debate about the interpretation of the probability concept.
One conception of probability is drawn from the idea of symmetrical
outcomes. For example, the two possible outcomes of tossing a fair coin seem not
to be distinguishable in any way that affects which side will land up or down.
Therefore the probability of heads is taken to be 1/2, as is the probability of tails.
In general, if there are N symmetrical outcomes, the probability of any given one
of them occurring is taken to be 1/N. Thus, if a six-sided die is rolled, the
probability of any one of the six sides coming up is 1/6.
Probabilities can also be thought of in terms of relative frequencies. If we
tossed a coin millions of times, we would expect the proportion of tosses that came
up heads to be pretty close to 1/2. As the number of tosses increases, the proportion
of heads approaches 1/2. Therefore, we can say that the probability of a head is 1/2.
If it has rained in Seattle on 62% of the last 100,000 days, then the
probability of it raining tomorrow might be taken to be 0.62. This is a natural idea
but nonetheless unreasonable if we have further information relevant to whether it
will rain tomorrow. For example, if tomorrow is August 1, a day of the year on
which it seldom rains in Seattle, we should only consider the percentage of the
time it rained on August 1. But even this is not enough since the probability of rain
on the next August 1 depends on the humidity. (The chances are higher in the
presence of high humidity.) So, we should consult only the prior occurrences of
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