Chance and the Sovereignty of God: A God-Centered Approach to Probability and Random Events by Vern S. Poythress

Author:Vern S. Poythress
Language: eng
Format: mobi, epub
Tags: Religion, Christian Theology, Apologetics
ISBN: 9781433536953
Publisher: Crossway
Published: 2014-04-28T22:00:00+00:00

CALCULATING CONDITIONAL PROBABILITY

Let us see how to reckon with conditional probability. We consider again the situation where Jill rolls a die, and it comes up 5. Jill then reveals to Karen that the result is odd.

Before Jill rolled the die, there were six possible outcomes. The probability of the result being 5 was 1 out of 6, or 1/6. What is the probability that the result would be odd? There are three cases of an odd outcome, namely 1, 3, and 5. These three cases are out of a total of 6 possible cases. The probability of odd is obtained by dividing the number of favorable cases (3) by the total number of cases (6). The probability is 3/6, which is the same as 1/2. Or we can obtain the same result by adding up the probability of the three distinct events, 1, 3, and 5. The probability of getting 1 is 1/6. Likewise, the probability of getting 3 is 1/6, and the probability of getting 5 is 1/6. Since these three outcomes are mutually exclusive, the probability of getting either 1 or 3 or 5 is 1/6 + 1/6 + 1/6 = 3/6 = 1/2.

Now look at things from Karen’s point of view. Jill has told her that the result is odd. Given this information (a condition), what is the probability that the outcome is 5? There are three possible odd outcomes, namely 1, 3, and 5. So from Karen’s point of view, the probability of getting 5 is 1 out of 3, or 1/3. This probability is a conditional probability, since it is dependent on the condition, known to Karen, that the outcome is odd.

We can now observe that there are two ways of obtaining the probability of the die coming up with 5 on top. The first way is to calculate it directly: there is one successful outcome out of a total of six possible outcomes, for a probability of 1/6. The other way is to do it in two stages. First, we observe that in order for the die to come up 5, it must come up odd. There are three ways to obtain this outcome. Given that it comes up odd, there is only one way out of three for it to come up 5. We move from a total of six outcomes at the beginning, to three outcomes that are odd, to one outcome with 5 on top. Getting 5 is one outcome out of 3 (the odd outcomes) out of 6. The probability of getting 5 is the probability of getting 5 out of the three odd outcomes, times the probability of getting an odd outcome, out of the 6 total possibilities. We can represent the reasoning compactly as follows:

1 outcome out of 3, once we have narrowed down to 3 outcomes out of 6.

By dividing by six, we can represent the same process in fractions or probabilities:

1/6 [1 outcome out of 6 where we get 5 on top] = 1/3 [1 out of 3 odd outcomes] × 3/6 [3 odd outcomes out of 6].

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