Probability - The Basic
Vuong Huynh
What is probability?
Definition: the likelihood of an event (outcome) occurring.
An event is likely or unlikely.
Use real number between 0 and 1 to represent a probability: 0.2, 0.5, ...
- 1: Absolute certainty of the event occurring.
- 0: Absolute certainty of the event NOT occurring.
Probability is a field of quantifying how likely each event is on its own.
- A -> event
- P(A) -> probability
- P(A) = preferred / all
(preferred or favorable is the outcome while all is the sample space)
Example in coin flip:
- A -> HEAD or TAIL - only preferred outcome
- Sample space is 2 (HEAD and TAIL)
- P(A) = 1/2 = 0.5
Example in rolling a dice:
- A -> only preferred outcome in {1, 2, 3, 4, 5, 6}
- Sample space is 6 (1 to 6)
- P(A) = 1/6 = 0.167
If we want the outcome when rolling dice is divisible by 3 (3 or 6)
- A = 2
- Sample space is still 6
- P(A) = 2/6 = 0.33
Remember
independent: P(A and B) = P(A) . P(B)
We express probabilities numerically: to compute which event is relatively more likely. Our goal is to be able to compare the probabilities of events and determine which is the relatively more likely outcome.
What is the probability of drawing a spade from a standard deck of playing cards?
We have 13 cards in the suit and 52 in the deck. When we apply the "favored overall" formula, we get P(spade) = 13/52 = 0.25.
Expected Values
Definition: the average outcome we expected if we run an experiment many times.
Flipping a coin multiple times:
- Trial -> flip and record outcomes
- Experiment -> multiple trials
Experimental Probabilities vs Theoretical probabilities
P(A) = successful trials / all trials
Expected value E(A) -> the outcome we expect to occur when we ren an experiment.
Catigorical Outcomes
E(A) = P(A) * n
Numerical Outcomes:
E(A, B, C) = A.P(A) + B.P(B) + C.P(C)
Which of the following classified as an experiment but not a trial?
Flipping a coin 20 times and recording the 20 different outcomes. we complete 20 different individual trials, which make up one experiment.
Why do we use experimental probabilities?
Because they are easy to compute and serve as good predictors for theoretical ones.
Why do we use intervals when forecasting future events?
Because the expected value might have a low probability or occur. We want to increase the likelihood of our predictions being accurate. The expected value could be unattainable.
Frequency
The probability frequency distribution is a collection of the probabilities for each possible outcome.
The experiment of rolling two dice and the variable is the sum of the outcomes in the dice. The expected value is synonymous with to mean of the distribution. Hence, the expected value is the mean of the sum of outcomes of two dice in the long run. In other words, if we roll two dice repeatedly, the sum of outcomes would be 7, on average.
In the example with the 2 standard six-sided dice, why is the probability of rolling a sum of 7 equal to one-sixth?
Because we have 36 possible outcomes when we throw the 2 dice and in 6 of those, we get a sum of 7. Thus, we have 6 favorable outcomes and 36 total possible outcomes and we plug these values in the formula to get 1 over 6.
What does the “frequency” of a value within the sample space represent?
The number of times the value features in the sample space.
Event and Their Complements
The complement of A is A'
- A -> rolling an even numbers
- A' -> NOT rolling an even number
- (A')' = A
Sum of all outcomes equal to 1.
- P(A) + P(B) + P(C) = 1
- A' = B + C
- P(A') = 1 - (A)
The complement of an event is everything the event is not.
The event and its complement add up the entire sample space (equal to 1).
The complement of HEADs is TAILs.