Probability and Counting Flashcards
What is the fundamental counting principal?
The total number of ways a task can be done is the product of the number of choices we have at each stage.
How do you calclulate the number of ways that N items can be arranges (example how many ways 6 books can be arranged)
N factorial: The product of N and every positive integer less than N
How do you approach counting problems with restrictions?
If a restriction appears in a counting problem that can be broken down into multiple stages always start with the most restrictive stage. If more than one stage has restrictions, then start at the most restrictive and then the second most, third most, and so on.
If the word NOT appears in a counting problem, it is a clue that you might want to consider using which strategy?
counting the number of arrangements that don’t obey the restriction and subtract them from the total
What should you consider if a counting question is asking about two terms being in a certain order and not another?
give an example
think about whether the groups are related by symmetry.
For example: if you are considering arranging 6 items and A must come at some point before B, this happens in exactly the same number of cases as B comes before A and thus the allowable arrangements would be half of the total number of possible arrangements.
How do you count the number of possible arrangements when you are dealing with identical items?
Calculate the total number of arrangements if all items were unique (i.e. the total number of items factorial)
Divide this by the number of ways the identical items can be arranged (i.e. the number of identical items factorial)
*I think this will only work when all items are included in the arrangement, does not work if you are choosing a smaller group of items to arrange from a larger pool that has identical items.
What is the formula for the total number of arrangements if there are multiple sets of identical items?
In what situation can this formula be used?
(b!)(c!)(d!)
where n = total number of items b = number of identical items in group b c = number of identical items in group c d = number of identical items in group d
*This formula can only be used when you are including all n numbers in the arrangement (this will not work if you are choosing items from a larger pool that contains repetitions this will not work)
What is nCr
the number of ways you can chose r individuals from a pool of n
nC1=
n
nCr=
nC(n-r)
this means that is we select a group of are from a pool of n, we actually create two groups, the group of r and the group of n-r
example: if we choose 4 from a group of 10, we are also creating a group of the 6 that are left behind. Since there is a corresponding group of 6 for every group of 4, there must the the name number of groups and therefore 10C4 = 10C6 (so 10C4 = 10C(10-4)
How do you solve nCr
start with the fundamental counting principal and then divide by r! to eliminate repetitions
Repetitions need to be eliminated in cases where order does not matter (e.g. picking a team of 3 from a group of 10).
nC2=
2
What do you use combinations?
If you are picking from a group with no repeats and order doesn’t matter
When do you use the fundamental counting principal?
When you’re picking from a group with no repeats and different orders of the final selection are meaningfully different (order matters)
What is probability?
probability is a ratio of the number of “successes” over the total number of outcomes
What is the compliment rule
P(not A) = 1 - P(A)
the probability of A NOT occurring is equal to 1 minus the probability of A occurring
In probability what is the simple AND rule? when does this apply?
P(A and B) = P(A)*P(B)
multiply (this is the simple and rule, only applies in situations where the events are independent)
In probability what is the simple OR rule? when does this apply?
P(A or B) = P(A)+P(B)
add (this is the simple or rule, only applies in situations where the events are mutually exclusive)
When are events mutually exclusive?
When it’s impossible for both of them to occur at the same time. In other words, if one occurs, it precludes the possibility of the other.
If two events (A and B) are mutually exclusive, what are the possible outcomes? what is nor possible?
1) A alone happens
2) B alone happens
3) Neither A nor B happen
*Impossible: A and B both happen
In probability, what is the General OR rule? when does this apply?
P(A or B) = P(A) + P(B) - P(A and B)
the probability of A or B equals the probability of A plus the probability of B minus the probability of A and B
This rule is ALWAYS true, but remember to apply it in situations where events are not mutually exclusive (i.e. they can occur together)
In probability, what is the General AND rule? when does this apply?
P(A and B) = P(A)*P(B I A)
the probability of A and B equals the probability of A times the probability of B given A
or the inverse
P(A and B) = P(B)*P(A I B)
This rule is ALWAYS true, but remember to apply it in situations where events are not independent (i.e. previous choice influence the probability of future choices)
If something is picked with replacement this means that ______________________
choices are independent: all choices are being made from the same pool and therefore each new choice is independent of previous choices
If something is picked without replacement this means that ______________________
choices are NOT independent: each choice is made under a different condition - each choice changes the probabilities of all subsequent choices
What does P (A I B) mean?
Conditional probability which means: what is the probability of A given B
In other words: assuming that we know the event B is true, what is the probability that A happens
When you see the words ‘at least’ in a probability question, what do you need to do?
Use the compliment rule as a shortcut
What is the probability of at least 1 success equal to?
1 - P(zero success)
1 minus the probability of zero successes
Give an example of mutually exclusive and independent
example of mutually exclusive would be heads and tails on a single coin flip (having heads means you can’t have tails and vice versa), however subsequent flips of a coin would be independent (the result of the first coin flip doesn’t influence the second and so on)
How do you use counting to calculate probability
count the total number of desired outcomes
OVER
the total number of possible outcomes
