Classical/theoretical probability

A set of equally-likely outcomes, the sample space S.
An event is any subset of S.
The probability of the event is the number of outcomes it has, i.e. the size of the event, divided by the total number of outcomes, i.e. the size of S. P(E)=|E|/|S|

Complement of event A is Ā = S-A. Probability of Ā = 1-P(A) and P(A)=1-P(Ā)

Probability experiment: roll a die

Sample space, S={1,2,3,4,5,6}     |S|=6
Some events:
The number is odd = E_odd={1,3,5}     |E_odd|=3     P(E_odd)=|E_odd| / |S| = 3/6 = .5
E_even={2,4,6}     Its probability is .5
E_<3={1,2}     P(this event)=2/6=.333
E_≥3={3,4,5,6} = complement of E_<3     Probability=4/6=.66666 = 1-P(complement)

E₇, rolling a 7={} the empty set. Probability is 0, impossible.
E₁₂₃₄₅₆, rolling a 1,2,3,4,5,or6={1,2,3,4,5,6}. Probability is 1, certain.

Probability experiment: roll two dice

Sample space S={

(1,1),	(1,2),	(1,3),	(1,4),	(1,5),	(1,6),
(2,1),	(2,2),	(2,3),	(2,4),	(2,5),	(2,6),
(3,1),	(3,2),	(3,3),	(3,4),	(3,5),	(3,6),
(4,1),	(4,2),	(4,3),	(4,4),	(4,5),	(4,6),
(5,1),	(5,2),	(5,3),	(5,4),	(5,5),	(5,6),
(6,1),	(6,2),	(6,3),	(6,4),	(6,5),	(6,6)}

|S| = 36
Some events:
Sum of dice is 2={(1,1)} snake eyes     P(this event)= 1/|S| = 1/36 = .02777
Sum of dice is 10={(4,6),(5,5),(6,4)}     Probability = ?
What is the probability of each of these events: sum is 2,3,4,5,6,7,8,9,10,11,12 ?
    So what is the most common sum when two dice are thrown?
What is the probability of the event "sum is less than 5": ?

Make a table showing the product of the two dice.
    What is/are the most common product(s)? and the probability ?
    What is the probability of getting a multiple of 6?

Probability experiment: roll three dice
What is the size of the sample space, |S|= ?
What is the probability of getting triples?
What is the probability of getting a sum of 5?
What is the probability of getting a sum ≤ 5?

Probability experiment: flip three coins (or flip one coin three times).

Sample space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}     |S|=8
Some events:
0 tails= {HHH}     Probability = 1/8=.125
1 tail= {HHT, HTH, THH}     Probability = 3/8=.375
2 tails= {HTT, THT, TTH}     Probability = 1/8=.375
3 tails= {TTT}     Probability = 1/8=.125
E_≥1H At least one head: {HHH, HHT, HTH, HTT, THH, THT, TTH}     Probability = 7/8=.875 =1-P(~E_≥1H) = 1-P(0 heads)
E_≥2H = {HHH, HHT, HTH, THH} = 0 or 1 tails
First is a head and third is a head= {HHH, HTH}
First is a head or third is a head= {HHH, HTH, HHT, HTT, TTH, THH}
First is a head xor third is a head= {HHT, HTT, TTH, THH}
The coins are all the same = {HHH,TTT}

Probability experiment: flip four coins (or flip one coin four times).
Sample space: ?
Some events:
Probabilities of 0, 1, 2, 3, 4 tails: ?