Midsemester Exam Flashcards
(25 cards)
Binomial Distribution
All trials are independent
Only 2 possible outcomes (Success or failures)
Probability of success is the same on every level
Usually replaceable items
Binomial Distribution Formula
P(x) = (nCx) Px Qn-x
n: number of trials, P: probability, Q: 1-probability, X: random variable
Shortcut Expected Value Formula
E(x) = np
n: number of trials, P: probability
Combination
A combination is an arrangement of objects where the order is NOT important.
There are three commonly used notations for combinations
nCr, C(n,r) and (n/r). The last is not a fraction!
Given abc:
Empirical Probability
A probability based on experimental results or a probability experiment.
Ex. Suppose you roll two dice 100 times and a sum of 10 comes up 7 times.
Then the Empirical probability of rolling a 10 is 7/10 or 7%.
Hypergeometric Distribution
A hypergeometric distribution is similar to a binomial distribution in that it has two possible outcomes, success or failure.
The hypergeometric distribution relies on dependent trials. The probability of success changes with each successive trial.
Formula: P(X=x)=(a/x)x(n-a/r-x)
/(n/r)
Intersection
The centre, where the two circles overlap is called the intersection.
The intersection represents the objects, or the number of objects, that belong to both sets.
Permutation
A permutation is an arrangement of objects in a specific order.
An arrangement of r objects from a total of n objects can be written as nPr.
If you have 12 books and you need to arrange 5 of them on the shelf, there are 95 040 different arrangements. You now have several ways to solve this problem. (Note – listing the possibilities or using a tree diagram will be so ineffective that they should not be considered as possible methods.)
Solution 1 – Using the Product Rule
12 × 11 × 10 × 9 × 8 = 95 040
Solution 2 –Using Permutation Notation 12P5
12P5 = 95 040
Probability
Probability is the chance that something will happen - how likely it is that some event will happen.
Probability Distribution
An example will make clear the relationship between random variables and probability distributions. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the variable X represent the number of Heads that result from this experiment. The variable X can take on the values 0, 1, or 2. In this example, X is a random variable; because its value is determined by the outcome of a statistical experiment.
A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. Consider the coin flip experiment described above. The table below, which associates each outcome with its probability, is an example of a probability distribution.
Number of heads Probability
0 0.25
1 0.50
2 0.25
The above table represents the probability distribution of the random variable X.
Subjective Probability
A subjective probability describes an individual’s personal judgement about how likely a particular event is to occur.
Theoretical Probability
Theoretical Probability is based on mathematical analysis of results.
Ex: Find the theoretical probability of drawing a ‘Get out of Jail Free’ Card.
There are 16 Chance cards and 16 Community Chest cards in the deck and 2 are ‘Get Out of Jail Free’ cards.
Then the probability of drawing this card is 2/32 or 0.0625%.
Note that theoretical probability is based on one trial. This calculation assumes that no cards have been removed from the deck.
Union
The union of two sets A and B is the set of elements in A or B (or both).
The symbol is a special “U” like this: ∪
Example: The union of the “Soccer” and “Tennis” sets is alex, hunter, casey, drew and jade
Soccer ∪ Tennis = {alex, hunter, casey, drew and jade}
Z-Score
Measures of standard deviation.
z = (X - μ) / σ
where z is the z-score, X is the value of the element, μ is the population mean, and σ is the standard deviation.
Product Rule
If there are B ways of doing something, and M ways of doing another thing after that, then there are n x m ways to perform both of these actions.
Look for the word AND
Sum Rule
If there are n ways of doings something, and m ways of doing another thing, both of which cannot be done at the same time, then there are n+m ways to choose one of these actions.
Look for the word OR
Factorial
A factorial is represented by the sign (!). When we encounter n! (known as ‘n factorial’) we say that a factorial is the product of all the whole numbers between 1 and n, where n must always be positive.
Ex. 1!= 1 = 1 2!= 2x1 = 2 3!= 3x2x1 = 6 4!= 4x3x2x1 = 24
0! is equal to 1.
Pascals Triangle
it’s symmetrical
potentially infinite in size
each number is the sum of the 2 numbers above it to the left and right
Combinations in the form C(row number, element number) also form Pascal’s Triangle
Pascal’s Identity: (n , r) = (n-1 , r-1) + (n-1 , r)
Discrete Data
Data is counted and can only take certain values.
Ex. the number of students in a class, number of threads in a sheet, number of kittens in a litter.
Continuos Data
Data is measured and can take any value (within a range).
Ex. a dog’s weight, a person’s height, the length of a leaf.
Dependent Event
If event B is dependent of A, then the conditional probability of B given A is described by the notation P(B | A).
If both events occur then probability is given by P(A and B) = P(A) × P(B | A).
Also, P(B | A) = P(A and B)/P(A)
Independent Event
Two events are Independent if the probability of event A has no impact on the probability of event B.
The probability of Independent events A and B both occurring is given by
P(A and B) = P(A) × P(B)
Mutually Exclusive Events
Lara always finds herself in Jail when she plays Monopoly®. What is the probability she will get out of jail?
Recall that to get out of jail – Lara must roll doubles OR draw a Get out of Jail Free Card. What is the probability that this happens?
The events required to get out of Jail are mutually exclusive. Lara can choose one option or the other in her attempt.
When two events are mutually exclusive – we ADD their probabilities.
P(Doubles or Get Out of Jail Card) = P(Doubles) + P(Get out of Jail Card).
**If event A happens, then event B cannot, or vice-versa.
Non-mutually Exclusive Events
Non-mutually exclusive events have elements in common.
P(A or B) = P(A) +P(B) - P(A and B)
P(A or B) are the events that are shared by both events. They are also called the INTERSECTION. The intersection is subtracted so that elements are not counted twice.
**Can’t happen at the same time, example: turning left and right.
Continuous Probability Distributions
a random variable that can assume all possible random
values (ie city temperature)
Probability Density Function: a function that describes how likely this random variable will occur at a given point.
Height formula: height = 1/(b-a) where b is the top range, and a is the bottom range given.
Normal Probability Distributions
used to solve continuous probabilities
symmetry about the mean
total area under the curve is 1
standard deviation is the distance from the mean to the point of inflection
Any normal distribution can be described as by the mean and the variance: so we often write N(mean, variance) to describe a distribution
The distribution chart shows area under the graph from the X value to the left end
Z-Scores can be calculated using Normal distributions:
Z = x – mean / standard deviation
Sometimes, you will have to subtract the mean to equalize. This makes it so the mean is on the center.
