Mobules 22&23: Combinatorics and Probability Flashcards
Combinatorics vs Permutations
Combinatorics: order doesn’t matter
Permutations: order DOES matter
Combination: Basic Combination Formula
nCk = n! / [(n-k)! * k!]
n = total number of objects k = number of objects that will be chosen
Combination: Box & Fill Method
1) Create X number of boxes for the amount of decisions that need to be made. This will be the numerator
2) Fill the box with the number of options available during each choice
3) Fill denominator with the factorial of the number of boxes
4) Solve
Fundamental Counting Principal
If there are M ways to perform task 1 and N ways to perform task 2 and the tasks are independent, there are m x n ways to perform both tasks together.
AND vs. OR
And = multiply the results Or = add the results
When I see I need to choose “at least” some number of items, I will
probably skip the question…
BUT if I don’t, need to find the combination value for each possible scenario included in the “at least.”
Collectively Exhaustive Scenarios
Two events are CE if together they represent all of the potential outcomes. (add all possibilites from each.)
When I see a “X items can’t be chosen together” problem, I will
1) Find the total possible ways if there were no restrictions.
2) Find the total possible ways that the two people could be chosen together.
3) Subtract: 1 - 2
Basic Permutation Formula
nPk = n! / (n - k)!
n = number of items k = number of objects chosen
Same as combination without the additional k! multiplied in the denominator.
When I see a combination/permutation problem, I will
BEFORE ANYTHING ELSE, determine if the order matters (permutation) or not (combination.) Different strategies, formulas, etc
Permutation: Box & Fill Method
1) Create X number of boxes for the amount of decisions that need to be made.
2) Multiply the values in the boxes. That’s it. (Combination requires a denominator of the # of boxes, factorial’d)
When I see a permutation problem, I will
be careful not to double-count indistinguishable permutations.
Permutation Formula for Indistinguishable Items
N! / r1! x r2! x r3!
Use R as each group of 2 or more indistinguishable items.
Permutations: If N Items are to be Arranged in a Circle (Formula)
(N - 1)!
When a permutation problem requires 2 items to be adjacent in the order, I will
consider them as one item. Then, to reflect the fact that they could be ordered 1-2 or 2-1, multiply the resulting number of combinations by 2.
Basic Probability Formula
number of desirable outcomes / number of total possible outcomes
When finding the probability of multiple events occurring, I will
determine if the events are independent (don’t affect the probability of one another) or dependent (they do.)
Probability of two non-mutually-exclusive events occuring
P(A) + P(B) - P(Both)
If there are X possible ways that an outcome can occur, and each way has the same probability, then the total probability of the situation is:
X * [probability of any one way]
When I see “at least” in a probability problem, I will
remember that each number above the “at least” threshold is mutually exclusive, and the probabilities of each need to be ADDED, not multiplied
When I See “at least ONE” in a probability problem, I will
use the 1 - x method to calculate the probability faster.
Trick to solve algebraic probability problems (probability is given but number of items is not)
Quadratic formula. First scenario, desired value is X, the total value is the total given. The second scenario, desired value is X - 1, total is total - 1.
