Probability & Arrangements

Prob & arrangements

Arrangements how items (ppl objects, digets…) can be arranged or lined up

Fundamental counting principal if event M can occur m ways & is followed by event N that occursrs n ways then M followed order, by N can occurs m*n ways

Factorial positive integers n, n!=n(n-1)(n-2)…321 used when repetitions are NOT allowed

(Special Case***) 0! = 1

ermutations all the possible arrangements of n items choosing r where order is important & order is important & NO repetitions nPr=P(n,r)=n!/(n-4) (A,B & B,A=2 set)

Combinations All the possible arrangements of n items choosing r where order is NOT important (A,B & B,A=1 set) nCr=C(n,r)=(n/r)=n!/(n-r)!r!

kw: group, committee, no distinguishing characteristics

NOTE: a permutation has more arrangements bc in permutations one pair in different orders=2 sets combinations the same pair = 1set

Intro to Probability

outcomes the possible results

ex) rolling a die {1,2,3,4,5,6}

Event a collection of 1+ outcomes

ex) rolling a 2 on a die

Sample Space the set of all possible outcomes

ex) rolling a die {1,2,3,4,5,6}

ex) flipping a coin {H,T)

Probability measure of likelihood/chance that an event will happen P(A)= # of times event A occurs/# of total possible outcomes

Types:

Experimental Conduct the trails, observe & record the data

ex) Flipping a coin 100 times, recording the results, getting a tail 65 times P(T)=65/100=.65=65%

theoretical finding the mathematical probability, what should happen over the long run

ex) rolling a 6 on a die P(R)=1/6=.167=16.7%

Geometric the ratio of the target area to the total area **(know are formls)

Complementary Events All the outcomes not in the event

Independent Events

Compound Events more than 1 event (kw- events A or / and B)

Independent Events “And” when the oucomes of 1 event has no affect o n the following event(s)

  1. events have nothing to do with eachother

  2. with replacement (problem must state that) P(A&B)=P(A intersection B) =P(A)*P(B)