Probability Flashcards
what is a random trial
process with two or more possible outcomes whose occurrence CANNOT be predicted with certainty
what are examples of random trials
rolling a dice
flipping a coin
random sample of population
event
any possible subset of ALL the possible outcomes of a random trial
what are the outcomes and events for a dice roll (
outcomes
- the possible numbers rolled (1, 2, 3, 4, 5, 6)
events include
- resulting exactly 6
- rolling all even
- rolling numbers add to 5
probability
the proportion of times an event would occur IF repeated random trials over and over IN THE SAME CONDITIONS
what is the notation for probability of event A
Pr[A]
what is the notation for probability of rolling a 5 on a dice
Pr[rolling a 5]
what must the probability for any event be
between 0 and 1 (inclusive)
how do venn diagrams represent probabilities visually
the area represents all possible outcomes of a random trial
mutually exclusive events
events that CANNOT occur at the same time
what are examples of mutually exclusive events
coin flip and dice rolls and having feathers + teeth
notation for “and” events
Pr[A and B]
what must mutually exclusive probabilities add to
ZERO
what kind of event does this show
mutually exclusive (no overlap between events)
probability distribution
a list of the probabilities of ALL mutually exclusive outcomes of a random trial
what kinds of values can a probability distribution be
discrete or continuous
discrete probability distribution
each outcome can only be defined by ONE value (like the combinations of two dice rolls adding to 7)
what is the probability of two dice rolls adding to 9? 7?
9
- there are four possible combinations to add to 9 and 36 total outcomes
4/36 = 0.11
7
- there are 6 possible combinations to add to 7 and 36 total outcomes
6/36 = 0.166
continuous probability distribution
there are almost infinite possibilities for values between two unique numbers because the variable can take any real number within that range
what is the probability of a variable in continuous probability distribution
basically zero
how do you find the probability of continuous data
on the curve, its the area under the graph
what kind of probability distribution is this
continuous probability distribution
what kind of probability distribution is this
discrete probability distribution
notation for mutually exclusive events (or events)
Pr[A or B] = Pr[A] + Pr[B]
what must the sum be for mutually exclusive events
1
how to find the probability of an event NOT occurring in mutually exclusive events
1 - probability of event occurring
what is the GENERAL addition rule
Pr[A or B] = Pr[A] + Pr[B] - Pr[A and B]
when are events independent
if knowing probability of one event DOES NOT affect the probability of another event
what is an example of independent probability
rolling two dice (because the number on one dice will NOT affect the number on the other dice)
multiplication rule
probability that both independent events occur = product of the probability of each event
when does this rule apply
for independent events
when does this rule apply
addition rule - for when events are mutually exclusive
what does this rule apply
for mutually exclusive events and trying to find the opposite of the probability A
what are events if they are NOT independent
dependent
what does dependence of events suggest
that the variables are associated
what type of variable serves as a null hypothesis
independent events/variables
OR probabilities imply
addition
when do we use addition
when events are MUTUALLY EXCLUSIVE
when do we use multiplication rule
when A and B are INDEPENDENT
what does AND imply
multiplication rule
how do you find the probability of an event multiple times
(say not rolling a 6 ten times in a row)
find the prob you want (not rolling 6 on a dice) and calculate it to the power of 1o (the number of times you are doing the trial in a row)
what are probability trees
a tool to assist in calculating the probability of events that consist of multiple random trials
when are probability trees helpful
when trying to ensure all possible sequences of events have been considered
probability that two children yields a boy and girl
there are two paths for the children being of different sexes
- having a boy than girl (prob is 0.250)
- having a girl than boy (0.250)
- given that events are mutually exclusive, add each probability
- there is a 0.500 chance of having two children boy and girl
dependent events
variables or events that are ASSOCIATED meaning they depend on each other to happen
conditional probability
the probability of that event occurring GIVEN THAT a condition is met
what does this notation mean
the probability of event A occurring given that the condition of B is met
what is the law of total probability
to find the overall probability of a particular event = sum its probability across EVERY possible condition weighted by the probability of that condition
general multiplication rule
Pr[A and B] = Pr[A] Pr{A|B] = Pr[B] Pr[A|B]
Bayes’ theorem
why is Bayes’ theorem useful
because you can use one conditional probability to determine the other
