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<h1>
Bayes' Theorem
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<time class="published" datetime="2018-11-10T12:29:00+01:00">
10 novembre 2018
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<li>Mathematics</li>
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<p>Let A and B be two events such that P(A|B) denotes the probability of the occurrence of A given that B has occurred and denotes the probability of the occurrence B of given that A has occurred, then:</p>
<div class="math">$$ P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$</div>
<div class="math">$$ P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^c)P(A^c)}$$</div>
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Premutations and combinations
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<p>Finding patterns in the possible ways events can occur is very useful in helping us count the number of desirable events in our sample space. Two of the easiest methods for doing this are with permutations (when order matters) and combinations (when order doesn't matter). </p>
<h2>Permutations</h2>
<p>An <strong>ordered</strong> arrangement r of objects from a set, A, of n objects (where <span class="math">\(0\lt r \leq n\)</span>) is called an r-element permutation of A. You can also think of this as a permutation of A's elements taken r at a time. The number of r-element permutations of an n-object set is denoted by the following formula: </p>
<div class="math">$$_{n}P_{r}=\frac{n!}{(n-r)!}$$</div>
<h2>Combinations</h2>
<p>An <strong>unordered</strong> arrangement of r objects from a set, A, of n objects (where <span class="math">\(0\lt r \leq n\)</span>) is called an r-element combination of A. You can also think of this as a combination of A's elements taken r at a time.</p>
<p>Because the only difference between permutations and combinations is that combinations are unordered, we can easily find the number of r-element combinations by dividing out the permutations (r!):
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<div class="math">$$_{n}C_{r}=\frac{_{n}P_{r}}{r!}=\frac{n!}{r!(n-r)!}$$</div>
<p>When we talk about combinations, we're talking about the number of subsets of size r that can be made from a set of size n. In fact, <span class="math">\(_{n}C_{r}\)</span> is often referred to as "n choose r", because it's counting the number of r-element combinations that can be chosen from a set of n elements.</p>
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Conditional Probability
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<p>This is defined as the probability of an event occurring, assuming that one or more other events have already occurred. Two events, A
and B are considered to be independent if event A has no effect on the probability of event B (i.e. P(B|A)=P(A)). </p>
<p>If events A and B are not independent, then we must consider the probability that both events occur. This can be referred to as the intersection of events A and B, defined as P(A∩B) = P(B|A)P(A). We can then use this definition to find the conditional probability by dividing the probability of the intersection of the two events (A∩B) by the probability of the event that is assumed to have already occurred (event A):</p>
<div class="math">$$ P(B|A)=\frac{P(A\cap B)}{P(A)}$$</div>
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