Probability: Sample Space

Sample space - the set of all possible outcomes of an experiment.

Tree Diagrams & Ordered Pairs are often used to list sample space.

Tree Diagram Example: Sketch the sample space for tossing 3 coins.

Ordered Pairs Example: Sample Space for rolling a pair of 6 sided dice.

The probability that a specific event will occur can be described as:

Example: The probability of drawing a black ball out of a bag of 13 coloured balls is 3/13. Find the probability of not drawing a black ball.

• Independent Events - 2 events occur so that neither one affects the probability of the other.

ex. Tossing a coin and rolling a die.

• Dependent Events - when 2 events occur that do affect the probability of the other.

ex. Drawing two cards from a deck without replacement.

Binomial Theorem

Pascal's Triangle [Magic 11's]

If a row is made into a single number by using each element as a digit of the number (carrying over when an element itself has more than one digit), the number is equal to 11 to the nth power or when n is the number of the row the multi-digit number was taken from.

COMBINATIONS

Permutations - Selecting and ordering (two actions)

Combunations - Selecting (one action)

Formula for combitations:

n!/r!(n-r)!

Ex.

10C5

1st step: Just like Permutation find n and r and arrange according to the formula

n=10

r=5

10!/5!(10-5)!

2nd step: Substract (n-r)!

10!/5!5!

3rd step: Evaluate

10*9*8*7*6*5!/5!5! -> cancel one 5! in numerator and denominator

4th step : simplify

10*9*8*7*6/5! --> 30240/120 = {252}

252 is the final answer

Ex. 2

A class that consists of 24 boys and 15 girls. Class committee must consists 10 students. How many ways can this be done if, there are to be 7 boys and 4 girls in this class committee

Calculate:

Formula: Boys * Girls

Boys

24C7

24!/7!(24-7)!

24!/7!17!

24*23*22*21*20*19*18*17/7!17!--->24*23*22*21*20*19*18/7!

1,744,364,160/7!

=364,104

Girls

15C4

15!/4!(15-4)!

15!/4!11!

15*14*13*12*11!/4!11!--->15*14*13*12/4!

32760/4!

=1,365

364,104*1,365 = {497,001,960}

Permutations with Case Restrictions

Permutations 1

Factorial Notation

Today we learned about factorial notation. The symbol n! (read as n factorial) means to multiply all of the positive integers from n all the way down to one (1).

FORMULA USED: n!= n(n-1)(n-2)...(3)(2)(1), where n is an element of the positive integers.

Example 4:

5!= ----->n=5 so when you plug in to the formula you get:

5(5-1)(5-2)(5-3)(5-4)

(5)(4)(3)(2)(1)

=120

We can remember these factorials:

• 0!= 1
• 1!= 1
• 2!= 2
• 3!= 6
• 4!= 24
• 5!= 120
• 6!= 720
• 7!= 5040
• 8!= 40320
• 9!= 362880
• 10!= 3628800

We can also simplify factorial equations:

Example 5:

a) 5!/4! ----->we can expand the 5! to (5)(4!)

(5)(4!)/4! ----->and the (4!)'s are able to cancel out each other leaving the (5)

=5

e) (s-2)!/(s+1)! ----->we can expand the (s+1)! into: (s+1)(s+0)(s-1)(s-2)!

(s-2)!/(s+1)(s+0)(s-1)(s-2)! -----> since there are two (s-2)!'s in both the numerator and the denominator, they can cancel out leaving us with the answer

=1/(s+1)(s+0)(s-1)