## 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".

## Counting Techniques

**Fundamental Principle of Counting**

If event *E*_{1} can have *n*_{1} different outcomes, event *E*_{2} can have *n*_{2} different outcomes, ..., and event *E _{m}* can have

*n*different outcomes, then it follows that the number of possible outcomes in which composite events

_{m}*E*

_{1},

*E*

_{2}, ...,

*E*can have is

_{m}*n*

_{1}×

*n*

_{2}× ... ×

*n*

_{m}We call this *The Multiplication Principle*.

## Number of Civil, Electrical, and Mechanical Engineers and Their Average Ages

**Problem**

In an organization there are CE’s, EE’s and ME’s. The sum of their ages is 2160; the average age is 36; the average age of CE’s and EE’s is 39; the average age of EE’s and ME’s is 32 and 8/11; the average age of the CE’s and ME’s is 36 and 2/3. If each CE had been 1 year older, each EE 6 years and each ME 7 years older, their average age would have been greater by 5 years. Find the number of CE, EE, and ME in the group and their average ages.

## A tank is supplied by two pipes A and B and emptied by a third pipe C

**Situation**

A tank is supplied by two pipes *A* and *B* and emptied by a third pipe *C*. If the tank is initially empty and all pipes are opened, the tank can be filled in 20 hours. If the tank is initially full and *A* and *C* are opened, the tank can be emptied in 4 hours. If the tank is initially full and *B* and *C* are opened, the tank can be emptied in 2 hours. Pipe *A* supplies 50 liters per minute more than *B*.

1. Find the rate of pipe *A* in liters per minute.

A. 120 | C. 110 |

B. 130 | D. 140 |

2. Find the rate of pipe *C* in liters per minute.

A. 170 | C. 150 |

B. 160 | D. 140 |

3. Find the capacity of the tank in liters.

A. 12,000 | C. 11,500 |

B. 12,500 | D. 13,000 |