Counting Methods and Probability Review Flashcards
Sets:
Are groups of values that half common property. The items in sets are called elements or members. The set with a no elements is called the empty set.
An important characteristic of sets is that elements are unique. They do not repeat. Order does not matter in the sets. {1, 2, 3} is the sane as {3, 2, 1}
Lists:
Order of elements matters, and duplication can occur. Order does matter. List elements can be uniquely identify by the position. Lists are not enclosed by { }.
Set Operations:
The intersection of sets A and B is written as A (upside U) B. The union of two sets is the set of all the elements that are elements of either or both sets and is written as A U B. If sets have no common elements, they are referred to as mutually exclusive, and their intersection is an empty set.
to find the number of elements in the union of two sets:
|A U B| = |A| + |B| - |intersection of A and B|
Probability of choices:
For mutually exclusive sets in which sets have no common elements:
“or” choices = addition
“and” choices = multiplication
A 3-letter password where no two letters can be the same: 26 x 25 x 24 = 15,600
A menu with 3 choices for soup and 4 salad option, and diner are permitted to select a soup or a salad: 3 + 4 = 7
The combinations formula:
n! / k! (n - k) !, where n is the number of items in a group as a whole, and k is the number of items in each subgroup formed. The ! symbol means factorial. Use when order does not matter.
Example: 5! = (5)(4)(3)(2)(1) = 120
Finding potential combination for multiple groups:
1) Find the number of combination for each group separately.
2) Multiple the combination of each group to get the total possible combination for multiple groups.
Find the number of possible sub group when choosing one item from the set. The number of possible sub groups will always equal the number of items in the set.
Restaurant A has five appetizer, 20 main courses, and 4 deserts. A meal consist of 1 appetizer, 1 main course and 1 desert. How many different meals can we order a restaurant A?
(5)(20)(4) = 400
Permutations differ from combinations in the permutations are ordered. By definition, each combination larger than 1 has multiple permutations. Indication of a permutation question are “how many ways/arrangements/orders/schedules are possible?”
First place winner receive a gold medal, second place receive silver metal, third place receive bronze and forth place receive a blue ribbon. If there are seven entrants in the contests, how many different arrangements of award winners are there? (7)(6)(5)(4) = 840
Probability that can be represented as a fraction, decimal, or percentage. Every probability is express as a number between 0 and 1 inclusive, with the probability of 0 meaning “no chance” and the probability of 1 meaning “guaranteed happen.” The higher the probability of the greater chance that the event will occur.
If all the possible outcomes of experiment are equally likely to occur, use this formula to calculate the probability:
Probability P(E) = number of desired outcomes/ number of possible outcomes
By the same logic , 1 - P(E) is the probability that the event will not occur.
To calculate the probability of two or more independent events occurring multiply the probabilities of the individual events.
Rolling a 3 four consecutive times on a six sided die: (1/6)(1/6)(1/6)(1/6) = 1/1296
A bag contains 10 marbles, with 4 blue and 6 red. If 2 marbles are removed without replacement, what is the probability that both marbles removed are red ?
1st probability = 6/10 = 3/5
2nd probability = 5/9
Total probability = (3/5)(5/9) = 3/9 = 1/3
A probability of one or another event occurring:
P(A or B) = P(A) + P(B) - P(A and B)
Events A and B are independent. P(A) is 0.60 and P(A or B) is 0.94. what is the probability that event B occurs?
Since events are independent, P(A and B) = P(A) x P(B).
0.94 = 0.60 + P(B) - 0.60P(B)
0.94 = 0.60 + 0.40P(B)
0.35 = 0.40P(B)
P(B) = 0.85
