## Poisson Probability Distribution

The number of occurrences in a given time interval or in a given space can be modeled using *Poisson Distribution* if the following conditions are being satisfied:

- The events occur at random.
- The events are independent from one another.
- The average rate of occurrences is constant.
- There are no simultaneous occurrences.

The Poisson distribution is defined as

where *x* is a discrete random variable

*P*(

*x*) = probability for

*x*occurrences

*μ*= the mean number of occurrences

## Probability: A Family of Five Children

**Problem**

A family chosen at random has 5 children. What is the probability that...

- all are girls or all are boys?
- there are 3 boys and 2 girls?
- at least 1 is a boy?

## 09 - Number of Leaps to Take to Catch the Lead

**Problem**

A cat takes 4 leaps to a dog's 3; but 2 of the dog's leaps are equivalent to 3 of the cat’s. The cat has a start of 50 leaps. How many leaps must the dog take to catch the cat?

## Number of Steps in the Escalator

**Problem**

A certain businessman, who is always in a hurry, walks up an ongoing escalator at the rate of one step per second. Twenty steps bring him to the top. Next day he goes up at two steps per second, reaching the top in 32 steps. How many steps are there in the escalator?

A. 80 | C. 50 |

B. 60 | D. 70 |

## Poker Hand: Probability that Five Cards are of the Same Suit

**Problem**

In a 5-card poker hand, what is the probability that all 5 are of the same suit?

## Probability Problems Involving Cards

A♥ | A♦ | A♣ | A♠ |

2♥ | 2♦ | 2♣ | 2♠ |

3♥ | 3♦ | 3♣ | 3♠ |

4♥ | 4♦ | 4♣ | 4♠ |

5♥ | 5♦ | 5♣ | 5♠ |

6♥ | 6♦ | 6♣ | 6♠ |

7♥ | 7♦ | 7♣ | 7♠ |

8♥ | 8♦ | 8♣ | 8♠ |

9♥ | 9♦ | 9♣ | 9♠ |

10♥ | 10♦ | 10♣ | 10♠ |

J♥ | J♦ | J♣ | J♠ |

Q♥ | Q♦ | Q♣ | Q♠ |

K♥ | K♦ | K♣ | K♠ |

## Probability

**Probability**

For outcomes that are equally likely to occur:

If the probability of an event to happen is *p* and the probability for it to fail is *q*, then

## Permutation Problems - 01

**Problem**

In how many ways can the letters of the word MATHALINO be arranged if the vowels are to come together?

**Problem**

In how many ways can the letters of the word MATHEMATICS be arranged if the consonants are to come together?

## Sets

**Definition**

A **set** is a collection of explicitly-defined distinct *elements*.

**Elements of a Set**

If 2, 4, 6, 8 are elements of *A*, we can then write

or it can be written using the **set-builder** notation

read as "*A* is the set of all *x* such that *x* is an even digit".