Chapter 3 Flashcards
(20 cards)
Probability experiment
-any experiment whose outcomes rely purely on chance
Ex: rolling a dice, flipping a coin
Simple event
-refers to a single specific outcome of probability experiment
Ex: getting a six from rolling a die
Flipping a coin and getting heads
Denote this as E
Sample space
-Collection of all possible simple events that can result from a probability, represents every possible outcome
Ex: rolling a die, sample space is {1,2,3,4,5,6}
Flipping a coin : S = {heads,tails}
Denoted by S
Probability distribution
- set of all possible outcomes and their corresponding probabilities, tells us how likely each possible outcome is
Ex: in a fair die roll, each outcome has a possibility of 1/6
Simple probability
-Likelihood that a specific event will occur, if all outcomes in the sample space are equally likely (like a fair die), then the probability of an event E is calculated by the formula P(E)=n(E)/n(S)
n(E) = number of favourable outcomes
n(S) total number of possible outcomes in the sample space
Ex: rolling affair die
S= {1,2,3,4,5,6}
Suppose the event E is getting an even number, the simple events that make up E are {2,4,6}
-there are three favourable outcomes and six possible outcomes in total
Therefore, the probability P(E) =n(E)/n(S) =3/6
Properties of simple probabilities
- The probability of an event is between zero and one: 0<P(E)<1
-this means the probability of an event cannot be negative or greater than one
-a probability of one means the event is guaranteed to occur
-a probability of zero means event is impossible - Probability of an event is the sum of probabilities of the simple events in it.
Ex: rolling a fair die with six sides, and the event were interested in is getting an even number
-Simple events that make up this event are {2,4,6} (these are even numbers on the die)
-each of these simple events has a probability of 1/6
-now they probability of getting an even number is just the sum of the probabilities of these individual events: P(even number) =P(2)+P(4)+P(6) = 1/6+1/6+1/6
- Probability of the sample space is 1: when conducting an experiment, we are guaranteed to observe 1 of the possible outcomes in the sample space
Properties of simple probabilities
- No event can have a probability less than zero or greater than one, if probability equals one event is certain to happen, if probability equals zero then event cannot happen
- If an event consists of several simple events, like getting an even number from a die, the probability of the event is the sum of the probabilities of the individual simple events
- The probability of all possible outcomes occurring in a sample space is always one because you’re guaranteed to get one of the outcomes when you conduct the experiment.
Empirical probability
-probability of an event based on the actual results of an experiment.
Formula: P(E)=f/n
f= number of times the event occurs in the experiment
n= total number of experiments
Ex: Eura die 30 times and you want to find the empirical probability of rolling a 6. You observed that the number six comes up 13 times out of the 30 rolls
P(getting a 6) = 13/30
-as the number of trials increase the empirical probability tends to get closer to the theoretical probability
Events
- Compliment of an event [NOT E]
-E does not occur, includes all outcomes that are not E (E^c) - Intersection of two events (E1 AND E2)
-Both E1 and E2 happen simultaneously.
Ex: if E1=1,2,3
E2=2,3,4
The intersection of E1&E2=2,3 - Union of two events [E1 OR E2]
-represents the event where either E1 or E2 or both happen
Ex: E1=1,2,3
E2=3,4,5
E1 OR E2 =1,2,3,4,5
-either one or the other event or both can happen
Mutually exclusive
-Two events are mutually exclusive if they cannot happen at the same time, events do not share any common outcomes
Ex: Event C: rolling a 2 on the die {2}
Event D: rolling a 5 on the die {5}
-these two events have no outcomes in common so they are mutually exclusive
-For mutually exclusive events E1 & E2, the probability of both events occurring is zero
Special edition rule for mutually exclusive events
-if events a and B are mutually exclusive (cannot happen at the same time) then
P(A OR B) = P(A)+P(B)
-no overlap between them
Ex: A: rolling a 1 or 2 P(A)=2/6
B: rolling a 3 or 4 P(B)=2/6
-these two events cannot happen at the same time so mutually exclusive
P(A OR B) = 2/6+2/6 =4/6
Complement rule
-probability of not E happening
-probability event E occurs plus probability that it does not occur must equal 1, since 1 of these must happen in any experiment
Ex: if the probability of raining tomorrow (E) is 0.3, than the probability that it does not rain (E^c):
P(E^c) = 1-P(E) =1-0.3=0.7
General addition rule
-if events A and B are any 2 events, the probability of either event A or event B occurring is:
P(A OR B)= P(A)+P(B)-P(A AND B)
Ex: drawing a card from a deck of 52 cards
A: drawing a heart P(A) = 13/52
B: drawing a queen P(B) = 4/52
-intersection of a and B is drawing the queen of hearts, which is one card in the deck so P(A AND B) = 1/52
P(A OR B) = 13/52 +4/52 -1/52 =16/52 is the probability of drawing either a heart or a queen or both
Contingency tables [Two way table]
-Used to organize and display bivariate [two variables] data
-shows how two categorical variables are related by listing one variable along the rows and the other along the columns
-each cell inside the table shows the number of observations or frequency that fit a particular combination of both variables
-individual cells are joint frequencies
-Row and column totals are marginal frequencies
Conditional probability
-probability of an event happening given that another event has already occurred [Probability of B given A]
Multiplication rule
-helps us calculate the probability of two events a and B happening together
Independence
- 2 events A and B are independent if the occurrence of one event does not affect the probability of another event
Ex: rolling a dye and flipping a coin, the outcome of one does not affect the other
Factorial [n]
-arranging n number of distinct objects.
-General case when you are arranging all of the objects
Permutations
-arrangement of a subset of K objects selected from a larger set of n distinct objects
-Not arranging all objects, just a subset of them.
“N permute k”
-order of selection matters
Combinations
-order of selected objects does not matter
-selecting key objects from a set of an objects for the order of selection does not matter
“n choose k”
