#MathsInMinutes Day: 33 Probability Theory
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Probability is the branch of mathematics that deals with measuring and predicting the likelihood of certain outcomes. It is both an application of set theory, and an entirely new theory in itself.
One way of looking at probabilities is by treating a range of possible results as elements of a set. Take foe example the case of a fair coin tossed three times. The set of all possible outcomes can be represented with elements consisting of three letters, one per coin toss, with H standing for heads and T for tails. Clearly this set has eight elements:
{TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}
Since one of these outcomes must occur, the sum of all these probabilities must be 1, and if the coin is fair and each outcome is equally probable, the likelihood of each case is 1/8.
More complicated questions about probabilities can be answered by considering specific outcomes as sets that are subsets of the previous set of all possible outcomes.
For example, we can see immediately that the set of outcomes with precisely two heads contains three elements, and so has a probability of 3/8.
But what about the probability that precisely one throw is a head given that at least one is a tail? If we know that at least one throw is a tail, we can restrict the set of outcomes to:
{TTT, TTH, THT, THH, HTT, HTH, HHT}
Two elements of this set, out of seven in total, have precisely one head – so the probability is 2/7.
Similar but more generalized arguments have allowed mathematicians to develop a set of axioms for probability, written in terms of the probability of sets and the operations defined on sets.
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