# 5.3 The Binomial Probability Distribution

While there are many discrete probability distributions, by far the most important one is the binomial distribution. To set the stage for a discussion of this distribution, we will need to cover two preliminary notions: (1) counting issues; and (2) the Bernoulli probability distribution.

Number of Prizes | Dollar Amount |

1 | 25,000 |

4 | 5,000 |

50 | 500 |

945 | 0 |

X | f(X) |

25,000 | 0.001 |

5,000 | 0.004 |

500 | 0.050 |

0 | 0.945 |

1.000 |

## 5.3.1 Counting Issues

We may generally view the concept of a permutation as an ordered set of objects. If any two of the objects are interchanged, we have a new permutation. For instance, the letters “a, b, c” form a particular permutation of the first three letters of the alphabet. If we interchange b and c, then “a, c, b” gives us another distinct permutation of these letters. More specifically, a permutation is any particular arrangement of r objects selected from a set of n distinct objects, r ≤ n.

What is the total number of permutations of r objects selected from a set of n distinct objects? To answer this, let us find

the number of permutations of n different objects taken r at a time, where

For instance, ...

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