Probability: A Family of Five Children

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

  1.   all are girls or all are boys?
  2.   there are 3 boys and 2 girls?
  3.   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♠

 

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.
 

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