Probability
The sample space of an experiment is the set of all possible
outcomes of the experiment. The outcomes of an experiment are
the elements of the sample space. The outcomes are disjoint (no
two outcomes can occur simultaneously). An event is any subset
of the sample space.
If an event can happen in a number of different ways (of which
only one can occur), the probability of its happening at all is
the sum of the several probabilities of its happening in the
several ways
E.g. suppose the experiment is flipping a coin 3 times
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
From our experience with flipping coins we know that the
outcomes are equally probable. We know, by definition, that the
sample space contains all outcomes, and since some one of the
outcomes must occur then we also know that the probability that
one outcome occurs is certainty. We arbitrarily define the
probability of a certain event as 1. Hence,
P(o1 or o2 or ... o8) = 1
P(o1 or o2 or ... o8) = P(o1) + P(o2) + ... + P(o8) Why?
P(oioj) = 0 for i  j
Look at one of our old friends:
|A or B| = |A| + |B| - |AB| and since AB is empty we have
|A or B| = |A| + |B|
Suppose that A and B are disjoint events.
P(A or B) = |A or B|/ |S| = (|A| + |B|) /|S| = |A|/|S| + |B|/|S|
= P(A) + P(B)
Since the elements of the sample space are equally probable, 8 x
P(o1) = 1 or P(o1) = 1/8
What
flip
What
P(T)
is the probability of flipping exactly 1 tails when you
1 coin?
is the size of the sample space? S = {H,T} and |S| = 2
= 1/2 = 0.5
Two events are independent if the probability of the occurrence
of one event does not affect the probability of the occurrence
of a second event.
Independent events are not disjoint events
P(e1e2) = P(e1)P(e2) where e1 and e2 are independent events
P(e1e2) = 0 where e1 and e2 are disjoint events
The probability of the occurrence of an impossible event is 0.
The probability of the occurrence of a certain event is 1
Conditional events are not independent as the probability of the
occurrence of event does affect the probability of the
occurrence of a second event
P(E) >= 0
P(S) = 1
P(e1 + e2) = P(e1) + P(e2) – P(e1e2)
P(E + Ec) = P(E) + P(Ec) = 1
Independence
P(e1e2) = P(e1)P(e2)
Conditional
P(e2|e1) = P(e2e1)/P(e1)
For independent events, P(e1|e2) = P(e1)
What is the probability of getting two tails if two coins are
tossed?
What is the probability of getting exactly two tails if three
coins are tossed?
What is the probability of getting exactly two tails if four
coins are tossed?
What is the probability of getting exactly 1 tail if four coins
are tossed?
What is the probability of getting exactly no tails if four
coins are tossed? Note that “no tails” is equivalent to 4 heads
What is the probability that six cards drawn at random from a
standard deck of fifty-two cards will be red cards?
An urn contains four white balls and five red balls. What is the
probability that a ball chosen from the urn is blue?
In a lottery, players win a large prize when they pick four
digits that match, in the correct order, four digits selected by
a random mechanical process. A smaller prize is won if only
three digits are matched. What is the probability that a player
wins the large prize? What is the probability that a player wins
the small prize?
There are many lotteries now that award enormous prizes to
people who correctly choose a set of six numbers out of the
first n positive integers, where n is usually between 30 and 60.
What is the probability that a person picks the correct six
numbers out of 40?
What is the probability that four cards dealt at random from an
ordinary deck of fifty-two cards will contain one card from each
suit? This problem is solved most easily using combinations but
is understood most easily using permutations!
When eight coins are tossed what is the probability of exactly
five heads? At least five heads?
A poker hand consists of five cards selected at random from a
standard fifty-two-card deck. Find the number of poker hands of
each of the following types.
(a) Flush: five cards of the same suit
(b) Straight flush: five consecutive cards from the same suit,
allowing an ace to be counted as either high (after king or
low as a 1)
(c) Four of a kind, plus a fifth card
(d) Full house: three of one kind and two of another
(e) Three of a kind, but not a full house or four of a kind
(f) Straight: five consecutive cards, not necessarily from one
suit
Suppose that someone flips two coins and one of the coins is
heads. What is the probability that the other coin is also
heads?
Imagine a couple with two children, each of whom is equally
likely to be a boy or a girl. Now suppose you are given the
information that one is a boy. What is the probability that the
other child is a boy?
What is the probability that the sum of two dice is seven?
What is the probability that the sum of two dice is seven given
that one die is 5?
What is the probability that the first die is five given that
the sum of the two dice is seven?
A pair of fair dice, one blue and one gray, are rolled. What is
the probability that the sum of the numbers showing face up is
eight given that both of the numbers are even?
An urn contains five blue and seven gray balls. Let us say that
two balls are chosen at random, one after the other, without
replacement.
(a) What is the probability that both balls are blue?
(b) What is the probability that the second ball is blue and
the first ball is not?
(c) What is the probability that the second ball is blue?
(d) What is the probability that at least one of the balls is
blue?
An urn contains five blue and seven gray balls. Let us say that
two balls are chosen at random, one after the other, with
replacement.
(e) What is the probability that both balls are blue?
(f) What is the probability that the second ball is blue and
the first ball is not?
(g) What is the probability that the second ball is blue?
(h) What is the probability that at least one of the balls is
blue?