Suppose that E and F are independent events in the
sample space S. Then, the following pairs of events are
also independent
E and F’ (state axioms)
Indepedent = disjoint events
E = (E intersect F) U (E intersect F’) - Disjoint Unions
P(E) = P(E intersect F) + P(E intersect F’) [additivity]
P(E intersect F’) = P(E) = P(E intersect F) = P(E) - (P(E) X P(F))
= P(E) X {1-P(F)}
= P(E) X P(F’)
An industrial plant obtains electricity from two identical
turbines.
A turbine of this kind has probability 0.1 of
breaking down in the course of a given week, and
breakdowns in the two turbines occur independently.
Find the probability that at least one of the turbines breaks down
next week.
Events E, F and G in the sample space S are said to be
independent if
they are pairwise independent (i.e. E, F independent;
E, G independent; F, G independent)
P(E intersect F intersect G) = P(E) X P(F) X P(G)
Three “fair” dice are rolled. Find the probability that: (a)
the same score is obtained on all 3 dice
S = {(x, y, z): x = 1, 2, .., 6; y = 1, 2, …, 6; z = 1, 2, …, 6}
The scores on the dice are independent, so the 6^3 = 216
outcomes in S are all equally likely, with probability 1/216
6 ways to get same score therefore 6 x 1/216 = 1/36
Three “fair” dice are rolled. Find the probability that:
b) different scores on all three dice
Let E2 = ‘different scores on all three dice’
Multiplication principle
6 different values of x
5 for y
4 for z
6 x 5 x 4 = 120
120/216
Three “fair” dice are rolled. Find the probability that: same score on 2 dice, different score on 3rd
E1, E2 and E3 are disjoint events with E1UE2UE3 = S
P(E1) + P(E2) + P(E3) = P(S) = 1
P(E3) = 1 -P(E1) - P(E2)
= 90/216 = 5/12
Let E, F and G be events in the sample space S. E and F
are said to be conditionally independent given G if:
Two “fair” dice are rolled. Let E = ‘score on 1st die is a 1’,
F = ‘score on 2nd die is a 6’. E and F are conditionally independent?
Two “fair” dice are rolled. Let E = ‘score on 1st die is a 1’,
F = ‘score on 2nd die is a 6’. E and F are independent.
suppose that E and F are disjoint events in the sample
space S, with non-zero probabilities. Then E and F cannot
be independent. WHY
