Basic Statistics

As the number of reperitions of a probability experiment increases, the proportion with which a certain outcome is observedgets closer to the probability of the outcome. 

any process with uncertain results that can be repeated.

S, a probability experiment is the collection of all possible outcomes

is any collection of outcomes from a probability experiment. An event consists of one outcome or more than one outcome. We will denote events with one outcome, sometimes called simple events,e_i. In general, events are denoted using capital letters such as E. 

1. The probability of any event E, P(E), must be greater than or equal to 0 and less than or equal to 1. That is \(≤\)1.

2. The sum of the probabilities of all outcomes must equal 1. That is, if the sample space S= {e₁, e₂,....,en}, then P(e₁) + P(e₂)+.....+P(e_n)=1

lists the possible outcomes of a probability experiment and each outcome's probability. 

the probability of the event is 0

the probability of the event is 1

an event that had a low probabilit of occuring.

The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment. 

P(E) ≈ relative frequency of E= frequency of E/ number of trials of experiment

when each outcomes has the same probability of occuring. 

If an experiment has n equally likely outcome and if the number of ways than an event E can occur is m, then the probability of E,P(E), is

P(E)= number of ways that E can occur = m

          number of possible outcomes          n

So if S is the sample space of this experiment

P(E) = N(E)/N(S)

when N(E) is the number of outcomes in E, and N(S) is the nuber of outcomes in the sample space.

Two events that have no outcomes in common

If E and F are disjoint (mutually exclusive) evenets, then

P(E or F)= P(E) +P(F)

For any two events E and F

P(E or F) = P(E) + P(F) - P(E and F)

Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted E^c, is all outcomes in the sample space S that are nor outcomes in the event E 

If E represents any even and Ec represents the complement of E, then

P(E^c)= 1-P(E)

Two events E and F are independent if the occurence of event E in a probability experiment does not affect the probability of event F

Two events are dependent if the occurence of event E in a probability experiment affects the probability of event F

If E and F are independent events, then

P(E and F)= P(E) * P(F)

If events E1, E2, E3,......En are independent, then

P(E₁ and E₂ and E₃, and ...... and E_n) = P(E₁) * P(E₂) *......P(E_n)

1. The probability of any event must be between 0 and 1, inclusive. If we let E denote any event, then 0 ≤ P(E) ≤ 1

2. The sum of the probabilities of all outcomes in the sample space mus equal 1. That is, if the sample space S= {e₁,e₂,.....e_n} , then P(e₁) + P(e₂) +..... + P(e_n)=1

3. If E and F are disjoint events, then P(E or F) = P(E) +P(F). If E and F are not disjoint events, then P(E or F)= P(E) +P(F)- P(Eand F)

4. IF E represents any event and E^c represents the complement of E, then P(E^c)= 1-P(E)

5. If E and F are independent events, then P(E and F)= P(E) * P(F)

The notation P(F|E) is read " the probability of event F given event E." It is the probability that the event F occurs, given that the event E has occured. 

If E and F are any two even then

P(F|E) = P(E and F)/ P(E)= N(E and F)/ N(E)

The probability of event F occuring, given the occurence of event E, is found by dividing the probability of E and F by the probability of E, or dividing the number of outcomes in E and F by the number of outcomes in E.

If a task consists of a sequence of choices in which there are p selections for the first choice, q selection for the second choice, r selections for the third choice, and so on, then the task of making these selection can be done in 

p*q*r*.....

different ways

if n ≥ 0 is an integer, the factorial symbol, n!, is defined as follows:

n!=n(n-1).....3*2*1

0!=1     1!=1

is an ordered arrangement in which r objects are chosen from n distinct (different) objects so that r ≤n and repetition is not allowed. The symbol _nP, represents the number of permutations of r objects selected from n objects.

the number of arrangements of r objects chosen from n objects, in which

1. the n objects are distinct,

2. repetition of objects is not allowed, and 

3. order is important

is given by formula 

_nP_r= n!/(n-r)!