Discrete Probability
Example 1: Sum of Dice
Suppose two dice are rolled to determine their sum.
What are the possible values for the sum?
What is the probability of getting each possible sum?
1
Discrete Probability
Example 2: Number of Heads
Suppose a fair coin is flipped 10 times. What are the
possible number of heads which result from this experiment?
What is the probability that no heads occur?
What is the likelihood that exactly 4 heads occur?
2
Discrete Probability
Setting for Discrete Probability
Let S = {E1, E2, E3, . . .} denote the set of possible
outcomes (sample points) from a random experiment and
let {p1, p2, p3, . . .} denote the associated probabilities;
that is, there is a chance of pk that outcome Ek occurs.
Definition 2.5: An event A in a discrete sample space
S is a collection of sample points; that is, any subset of
S.
Requirements of Probability Measure
Axiom 1: P (A) ≥ 0 for every event A;
Axiom 2: P (S) = 1;
Axiom 3: If A1, A2, A3, . . . form a sequence of pairwise
mutually exclusive events in S, then
P (A1 ∪ A2 ∪ A3 ∪ · · ·) =
3
∞
X
i=1
P (Ai).
Discrete Probability
Example 1 revisited: Sum of Dice
Suppose two dice are rolled to determine their sum.
The probability distribution (or probability measure) of
the sum is
Sum 2 3 4 5 6 7 8 9 10 11 12
Prob.
1 2 3 4 5 6 5 4
36 36 36 36 36 36 36 36
3
36
2
36
1
36
Example 2 revisited: Number of Heads
Suppose a fair coin is flipped 10 times. What is the
probability distribution for the number of times the coin
lands heads?
4
A Question of Modelling
Example 1 revisited: Sum of Dice
Suppose two dice are rolled to determine their sum and
the probability distribution of the sum is
Sum
2 3 4 5 6 7 8 9 10 11 12
Prob.
1 1 1 1 1 26 1 1
36 36 36 36 36 36 36 36
What can we say about the dice?
5
1
36
1
36
1
36
First Problem
In the 1600s, there were two popular dice games which
gamblers played.
The first game consisted of rolling a die 4 times. The
player would win the amount of his bet if at least one of
the 4 rolls resulted in a six.
The second game involved rolling two dice 24 times. A
player would win the amount of his bet if at least one of
the 24 rolls resulted in a pair of sixes.
The understanding at the time was that these two games
had the same chance of success. However, the Chevalier de Méré, apparently based on his success in playing
the games, questioned whether the games had the same
chances of success and asked Pascal to study the matter.
This initiated the study of probability.
What are the chances of the player losing in each game?
6