Probability MATH 152

Experimental Probability

The probability of a event occurring based on the results of experiment and trials.

probability = # of occurrance / total # of trials.

The more trials you do the closer the experinment probability is liekly to be the theoretical probability.

Theoretrical Probability

The probability of an event calculated based on the possible outcomes in a sample space, without conducting experiments. It is expressed as the ratio of the number of favorable outcomes to the total number of possible outcomes.

probability using sets

A method for calculating probabilities by analyzing the relationships and intersections of different sets, often using Venn diagrams. This approach helps to visualize and compute the likelihood of events occurring together or separately.

union of sets

The combination of two or more sets, including all elements that are in any of the sets. In probability, the union represents the likelihood of at least one of the events occurring.

It is denoted as A ∪ B for sets A and B.

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
to find the probability you do n(A ∪ B)/n(S).

Conditional probability

The probability of an event occurring given that another event has already occurred. It is calculated using the formula P(A|B) = P(A ∩ B) / P(B), where A and B are two events.

multiplication law

The rule that states the probability of the occurrence of two independent events is the product of their individual probabilities. It is expressed as P(A and B) = P(A) × P(B) for independent events A and B.

Dependent events

Events where the occurrence of one event affects the probability of the other event. For dependent events A and B, P(A|B) is not equal to P(A).

P(A and B) = P(A) * P(B given A)

permutations

number of ordered arrangement of objects. The number of permutations of n object is n!

Ex. How many different orders can A B and C be arrange in?

3 ordered arrangement so its 3! or 6 is the number of possible order. if r is equal to n

r is the amount of object taken from n objects

The number of ordered arrangments of n items taken r at a time is P(n,r) n!/(n-r)!

binomial coefficient

counts the number of ways to form an unordered collection of k items chosen from a collection of n distinct items.

C(n,r) = n!/(n-k)!k!

Bimonial Probability Distribution

Describes the probability of the number of success in an experinment.
Fixed # of trials
2 possible outcomes for each trial

Constant probability of success/independent trials.

The probability of have k successes in n trials:

P(X = k) = n!/(n-k)!k! * p^k (1-p)^(n-k)
p is the probability of success and 1-p is the probability of failure.

n - k is the number of failures.

Geometric probability distribution

Models the probability of the # of trials nedded to achieve the 1st success in an experinment
2 possible outcomes for each trial
constant probability of success

Independent Trials

P(X = k) = (1-p)^(k-1) * p
X = number of trial till first success.

Complementary Probability

refers to the probability of the opposite event happening. In other words, if you know the probability of an event occurring, the probability of its complement (the event not occurring) can be found by subtracting the probability of the event from 1.

Law of Large Numbers

The Law of Large Numbers states that as the number of trials in an experiment increases, the average of the results will get closer to the expected value or true probability of the event. In other words, the larger the sample size, the more the observed average will reflect the theoretical average.