Step By Step Calculus » 8.4 - Equally Likely Conditional Probability

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Synopsis
Random Experiment: A random experiment is an experiment whose outcome can not be predicted with certainty.
Outcome/Sample Point: An outcome or a sample point is a possible outcome of a random experiment. For example, ‘H’ is a sample point in the coin toss experiment.
Sample Space: A sample space is the set of all possible outcomes in a random experiment. For example, \{H, T\}\{H, T\} is the sample space in the coin toss experiment.
Event: An event is a set of zero or more possible outcomes. In other words a subset of the sample space.
Equally Likely Conditional Probability: The probability of event EE given event FF when all outcomes in FF are equally likely is
\displaystyle P(E|F)=\frac{\#(EF)}{\#(F)},
\displaystyle P(E|F)=\frac{\#(EF)}{\#(F)},
where \#(EF)\#(EF) and \#(F)\#(F) denote the number of points in EFEF and FF, respectively.
Principle of Counting (PC): The Principle of Counting (PC) says that if we have rr experiments, and if the i^{\textrm{th}}i^{\textrm{th}} experiment has n_{i}n_{i} possible outcomes, then there are n_{1}n_{2}...n_{r}n_{1}n_{2}...n_{r} possible outcomes for the combined experiment.
Classification: If AA and A_{1},\dots,A_{r}A_{1},\dots,A_{r} are sets of possible outcomes such that any outcome in AA is in exactly one of A_iA_i, then
\displaystyle \displaystyle # of outcomes in AA\displaystyle =\displaystyle \sum_{i=1}^{r}\displaystyle =\displaystyle \sum_{i=1}^{r}# outcomes in A_{i}A_{i}\displaystyle . \displaystyle .