Module 1: Probaability Flashcards
Random experiment
procedure whose outcome in a particular performance cannot be predetermined.
- Must be able to list the set of all possible outcomes
- In general, random experiments must be able capable in theory of indefinite repetition
- It must also be possible to observe the outcome of each repetition of the experiment
Trial
Each repetition of the procedure for the random experiment and gives rise to one possible outcome only
Sample space
Set of all possible outcomes in a random experiment
An Event
Any subset of the sample space of S
Pr(an Outcome)
Pr(outcome) = Number of ways of getting that outcome / number of possible outcomes
What is the probability of getting a 6 when you roll a die?
Odds of getting a 6?
Pr(6) = 1/6
Odds of a 6: 1:5
What is a fair game
No one is expected to lose and no one is expected to win in the long run
Payback
Win percentage
= total payout for a winning number / amount bet across all numbers
Fair payout when you pay R1, 6 times?
R6 but the firm wont make profit so they can pay out R5.50 instead
Win percentage and house advantage when you payout R5.50 and retain 50c?
WP = 5.50/6 * 100% = 91.67
Therefore,
House advantage = 8%
Win percentage in a non fair game
WP = Pr(win)(Payout-bet) - Pr(lose)(bet)
Example:
A game consists of tossing a fair die of 6 and observing the absolute difference between the 2 outcomes. If its 3 or more, then you will double your money back. If not you lose your money.
- You pay R2 to play
What is the house advantage?
whoa is the fair payout?
Pr(0)=6/36 Pr(1)=10/36 Pr(2)- 6/36 Pr(3)=6/36 Pr(4)=4/36 Pr(5)=2/36 Pr(6)=2/36
Pr(outcome=>3) = 12/36
(1/3) of your time you expect to win R4
(2/3) you expect to lose (R2)
(1/3)(4-2)+(2/3)(-2_= -33.33% of R2
Therefore house advantage = 33.33%
Fair payout:
(1/3)(X-2)+(2/3)(-2)=0
X = 6
How to calculate fair payout
Pr(win)(X-bet) + Pr(lose)(0-bet)=0
What is a Set A?
A collection of distinguishable objects or entities.
A={s,d,e}
What are s,d and e?
Elements
What is are elementary events?
Event with exactly one member
Events D = {3} and E = {5}
What is G C H?
H C G?
G is a subset of H
H contains G
What is an intersection?
The set containing those elements which are common to both.
Can be an empty set - these are mutually exclusive or disjoint.
Union
The set that contains the elements that belong to either A or to B or to both A and B
Complement
A set of elements which are not in a particular set
I.e
__
A
___
Pr(A) =
1 - Pr(A)
___
Pr(A n B) =
Pr(A) - Pr(A n B)
Pr(A u B)
Pr(A) - Pr(B)
Number of distinguishable arrangements of n distinct objects not allowing repetition?
I.e A = {1,2,3}
N!
= 3! = 6
Order matters
Repetition isn’t allowed
P
