Say that in a particular city, 48% of homes have broadband internet installed, and 6 % of homes have both cable television

*and*broadband internet (we are obviously talking about a second-world country).

The question is: what is the probability that a particular home has cable TV,

*given that*it has broadband?

If X and Y are events, we write the

*conditional probability*of X given Y as P(X/Y).

Mathematically, this is defined as follows;

P(X/Y) = P(X & Y) / P(Y)

(This only makes sense when P(Y) does not equal zero).

In the above example, we take X to be the event that the house has cable TV, and Y to be the event that it has broadband. Notice that we do not have to know P(X) to calculate the answer:

P(X/Y) - 0.06/0.48 = 0.125, or 12.5 %.

In many contexts, conditional probability is extremely useful, as it allows probabilities to be updated as new information becomes available.

This is known as

*Bayesnian inference*.

In 1794, an important paper by the reverend Thomas Bayes was published posthumously. In it he gives a compelling account of conditional probabilities.

**The basis is**

*Bayes' Theoerem*, which states that for any events X and Y:**P(X/Y) = P(Y/X) x P(X)/P(Y)**

In a sense, this formula is not deep. It follows directly from the definition of conditional probability:

P(Y/X) = P(X & Y)/P(X)

so

P(X & Y) = P(Y/X)P(X)

Substituting this into the definition of P(X/Y) produces the result.

However, this theorem has been of great use, for example in the analysis of

**the problem of false positives.**

From: Mathematics 1001, by Dr. Richard Elwes

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