Counting and Probability Flashcards
What is the basic counting principle?
If there are n₁ ways to do one thing (outcomes of one experiment) and n₂ ways to do another thing (outcomes of another experiment), then there are n₁ * n₂ ways to do both.
What can the basic counting principle be generalized as, in terms of k number of experiments?
The total number of outcomes of k experiments is: n₁ * n₂ * n₃ * ——– * nₖ
What is the formula for arranging n number of distinct objects in n slots, -where ordering matters-?
(n)(n - 1)(n - 2)(n - 3) ————— (2)(1) = n!
What is the formula for the number of ways of filling n slots with n objects, where ordering matters, if n₁ of them are of the first kind, n₂ are of the second kind, and n₃ are of the third kind?
n!/[(n₁!)(n₂!)(n₃!)]
How do you calculate the number of outcomes with restrictions on the selections?
First fill the boxes that come with the restriction by the number of outcomes, before filling the other boxes with the number of possible outcomes.
What are permutations without repetition?
The various ways in which r number of objects (a subset) may be selected from n number of distinct objects (a set) without replacement (without repetition) and where ordering matters.
What is the formula for permutations without repetition?
n!/(n - r)!
What are combinations without repetition?
The various ways in which r number of objects (a subset) may be selected from n number of distinct objects (a set) without replacement (without repetition) and where ordering does not matter.
What is the formula for combinations without repetition?
n!/r!(n - r)!
What is probability?
Probability is a number estimating the likeliness of an event happening.
Does probability tell us exactly what will happen?
No, it is just a guide for what may happen.
What is the formula for the probability of an event happening?
favorable outcomes / total number of possible outcomes
What does the notation P(A) mean?
It translates to “Probability of Event A”
If A is the event and A’ is the complimentary event, then:
P(A’) = 1 - P(A) OR P(A) + P(A’) = 1
What is the formula for the number of expected favorable outcomes of an event?
P(E) * # of attempts
How can odds be expressed?
As a fraction or as a ratio, such as 5:1, or 5 to 1.
What is the formula for odds in favor of an event?
number of favorable outcomes / number of unfavorable outcomes
What is the formula for odds against an event?
number of unfavorable outcomes / number of favorable outcomes
After dividing both the numerator and denominator by the total number of possible outcomes of the experiment, odds in favor of an event (E) = …
P(E)/P(E’) = P(E)/[1 - P(E)]
If Events A and B are independent (the outcome of Event A does not affect the outcome of Event B), what is the formula for P(A and B)?
P(A) * P(B)
If Events A and B are dependent (the outcome of Event A affects the outcome of Event B), what is the formula for P(A and B)?
P(A) * P(B/A)
What is the formula for the experimental probability of an event?
number of times the event occurs / number of times the experiment is repeated
What is the formula for geometric probability?
P(a point chosen at random is from a particular region) = area of a particular region / total possible area
