Mach 9.3 Probability -- Sage

A sample space is a set S of all possible outcomes O of an experiment.

An event E is a subset of a sample space. So E is also a set of outcomes.

The probability P of an event E is: P(E) = \frac{n(E)}{n(S)}.

What is the sample space S for throwing 2 dice?

S = [(die1, die2) for die1 in [1..6] for die2 in [1..6]] S

len(S)

What is the event space E for the sum of the dice is 7?

E = [throw for throw in S if sum(throw) == 7] E

len(E)

When throwing two dice, what is P( sum of 7)?

e = Integer(len(E)) s = Integer(len(S)) e/s

def P(E, S): return Integer(len(E))/Integer(len(S))

bothprime = [(a,b) for (a,b) in S if is_prime(a) and is_prime(b)] bothprime; P(bothprime, S)

Probability Function

A probability function P assigns a real number to each outcome O in a sample space S such that:

0 \le P(O) \le 1 for every O in S.
\sum _{i = 1}^{n} P(O_i) = 1
P(\emptyset) = 0

[P([O], S) for O in S]

sum(_)

sums = [sum(O) for O in S] [(s, sums.count(s)) for s in [2..12]]

A = [O for O in S if is_even(sum(O))] B = [O for O in S if is_prime(sum(O))]

A; len(A)

P(A,S)

B; len(B)

P(B,S)

Conditional Probability

P(A | B) = \frac{P(A \cap B)}{P(B)}

def intersect(A, B): return [O for O in S if O in A and O in B]

intersect(A,B)

P(intersect(A,B),S)

P(intersect(A,B),B)

Multiplication Principle

P(A \cap B) = P(A) \cdot P(B | A)

P(A,S)*P(intersect(B,A),A)

P(intersect(A,B),S)

Addition Principle

P(A \cup B) = P(A) + P(B) - P(A \cap B)

def unite(A, B): return [O for O in S if O in A or O in B]

unite(A, B)

P(unite(A,B),S)

P(A,S) + P(B,S)

P(A,S) + P(B,S) - P(intersect(A,B),S)

Binomial Distribution

\sum _{r = 0} ^{n} {n \choose r}p^{n-r}q^{r} where p + q = n.

#var('p q') p = 1/4 q = 1-p n = 10 [binomial(n,r)*p^(n-r)*q^r for r in [0..n]]

[x.n(digits = 5) for x in _]