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♠

 

Number of Marbles in a Bag Containing Black and White Marbles

Problem
If 2 marbles are removed at random from a bag containing black and white marbles, the chance that they are both white is 1/3. If 3 are removed at random, the chance that they all are white is 1/6. How many black marbles are there?
 

Answer Key

 

Probability

Probability
For outcomes that are equally likely to occur:

$P = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

 

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

$p + q = 1$

 

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

$A = \{0, ~ 2, ~ 4, ~ 6, ~ 8\}$

or it can be written using the set-builder notation

$A = \{x ~ : ~ x \text{ is an even digit} \}$

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

Counting Techniques

Fundamental Principle of Counting
If event E1 can have n1 different outcomes, event E2 can have n2 different outcomes, ..., and event Em can have nm different outcomes, then it follows that the number of possible outcomes in which composite events E1, E2, ..., Em can have is

n1 × n2 × ... × nm

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.
 

Quiz: Random Problems 05

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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

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