Probability and Counting

Question 1

What is the fundamental counting principal?

Answer 1

The total number of ways a task can be done is the product of the number of choices we have at each stage.

Question 2

How do you calclulate the number of ways that N items can be arranges (example how many ways 6 books can be arranged)

Answer 2

N factorial: The product of N and every positive integer less than N

Question 3

How do you approach counting problems with restrictions?

Answer 3

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.

Question 4

If the word NOT appears in a counting problem, it is a clue that you might want to consider using which strategy?

Answer 4

counting the number of arrangements that don’t obey the restriction and subtract them from the total

Question 5

What should you consider if a counting question is asking about two terms being in a certain order and not another?

give an example

Answer 5

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.

Question 6

How do you count the number of possible arrangements when you are dealing with identical items?

Answer 6

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.

Question 7

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?

Answer 7

(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)

Question 8

What is nCr

Answer 8

the number of ways you can chose r individuals from a pool of n

Question 9

nC1=

Answer 9

n

Question 10

nCr=

Answer 10

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)

Question 11

How do you solve nCr

Answer 11

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).

Question 12

nC2=

Answer 12

2

Question 13

What do you use combinations?

Answer 13

If you are picking from a group with no repeats and order doesn’t matter

Question 14

When do you use the fundamental counting principal?

Answer 14

When you’re picking from a group with no repeats and different orders of the final selection are meaningfully different (order matters)

Question 15

What is probability?

Answer 15

probability is a ratio of the number of “successes” over the total number of outcomes

Question 16

What is the compliment rule

Answer 16

P(not A) = 1 - P(A)

the probability of A NOT occurring is equal to 1 minus the probability of A occurring

Question 17

In probability what is the simple AND rule? when does this apply?

Answer 17

P(A and B) = P(A)*P(B)

multiply (this is the simple and rule, only applies in situations where the events are independent)

Question 18

In probability what is the simple OR rule? when does this apply?

Answer 18

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)

Question 19

When are events mutually exclusive?

Answer 19

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.

Question 20

If two events (A and B) are mutually exclusive, what are the possible outcomes? what is nor possible?

Answer 20

1) A alone happens
2) B alone happens
3) Neither A nor B happen

*Impossible: A and B both happen

Question 21

In probability, what is the General OR rule? when does this apply?

Answer 21

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)

Question 22

In probability, what is the General AND rule? when does this apply?

Answer 22

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)

Question 23

If something is picked with replacement this means that ______________________

Answer 23

choices are independent: all choices are being made from the same pool and therefore each new choice is independent of previous choices

Question 24

If something is picked without replacement this means that ______________________

Answer 24

choices are NOT independent: each choice is made under a different condition - each choice changes the probabilities of all subsequent choices

Question 25

What does P (A I B) mean?

Answer 25

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

Question 26

When you see the words ‘at least’ in a probability question, what do you need to do?

Answer 26

Use the compliment rule as a shortcut

Question 27

What is the probability of at least 1 success equal to?

Answer 27

1 - P(zero success)

1 minus the probability of zero successes

Question 28

Give an example of mutually exclusive and independent

Answer 28

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)

Question 29

How do you use counting to calculate probability

Answer 29

count the total number of desired outcomes
OVER
the total number of possible outcomes