1. Algorithms CSCI 235, Fall 2017 Lecture 10 Probability II

2. Conditional Probability
Conditional probability (having prior knowledge):
Probability of A given B =
Example: Experiment A with H1 = 1/3, H2 = 1/4, H3 = 1/5
What is the probability of A3 (exactly 2 tails) given A4 (two consecutive flips the same)?
A3 = {HTT, THT, TTH}
A4 = {HHH, HHT, HTT, THH, TTH, TTT}

3. Example 2
Ann has 2 children, one of which is a girl. What is the probability that the other child is a boy? (Assume a child is a girl or boy with equal probability)
Pr{G} = 1/2 Pr{B} = 1/2

4. Independence If two events are independent then
Probability of A given B =
(Knowledge of B does not change the probability of A)
Therefore:
Definition: Events A and B are independent if

5. Example of independence
Of the four events in experiment A, which are independent?
Recall:
Event A1: First flip is head = {HHH, HHT, HTH, HTT}
Event A2: Second flip is tail = {HTH, HTT, TTH, TTT}
Event A3: Exactly 2 tails = {HTT, THT, TTH}
Event A4: Two consecutive flips are the same=
{HHH, HHT, HTT, THH, TTH, TTT}

6. Checking for independence
1. Are A1 and A2 independent?
Pr{A1}=Pr{HHH} + Pr{HHT} + Pr{HTH} + Pr{HTT}
= 1/60 + 1/15 + 1/20 + 1/5 = 20/60 = 1/3
Pr{A2}=Pr{HTH} + Pr{HTT} + Pr{TTH} + Pr{TTT}
= 1/20 + 1/5 + 1/10 + 2/5 = 3/4
If A1 and A2 are independent, then
= 1/20 + 1/5 = 5/20 = 1/4
Pr{A1}Pr{A2} = (1/3)(3/4) = 1/4 Yes. They are independent.
2. Are A1 and A3 independent? (We will work this out in class)

7. Discrete Random Variables
A discrete random variable maps (assigns) a real number to each element (outcome) of a sample space, S.
Example: h(x) = number of heads in outcome x
r(x) = length of longest run in outcome x
Outcome(x) h(x) r(x)
H1H2H3 3 3
H1H2T3 2 2
H1T2H3 2 1
H1T2T3 1 2
T1H2H3 2 2
T1H2T3 1 1
T1T2H3 1 2
T1T2T3 0 3

8. The Pre-Image
The pre-image of a value, v, under a discrete random variable f is
defined as:
In other words: the pre-image of v under f is the set of all outcomes for which the value of f(x) = v.
Example:
h-1(1) = {HTT, THT, TTH}
r-1(3) = ?

9. The probability density function
The probability density function (PDF) of a discrete random variable is defined as follows:
In other words: The probability density function of f for a value, v, is the sum of the probabilities of all the outcomes in the pre-image of v under f.
Example: PDF(h) = ?
(PDF(h))(0) = Pr{h-1(0)} =
Pr{TTT} = ?
(PDF(h))(1) = Pr{h-1(1)} =
Pr{HTT, THT, TTH}=?

10. Independence of discrete random variables
Discrete random variables f and g are independent if, for all v and w, f-1(v) and g-1(w) are independent.
Example: Are h and r independent?
Test for v = 0, w = 1: is h-1(0) independent of r-1(1)?
h-1(0) = {TTT} r-1(1) = {THT, HTH}

11. Expected Value
The expected value, E(f), of a discrete random variable, f, is a weighted average:
Example: What is the expected number of heads in Experiment A?
= 0(Pr{h-1(0)} + 1(Pr{h-1(1)} + 2(Pr{h-1(2)} + 3(Pr{h-1(3)}
= 13/30 + 2(9/60) + 3/60 = 47/60

12. Example 2
What is the expected number of flips in Experiment B (flip until heads)? Assume a fair coin.
Outcome b
H 1 b-1(1) = H
TH 2 b-1(2) = TH
TTH 3 b-1(3) = TTH
Pr{b-1(1)} = Pr{H} = 1/2
Pr{b-1(2)} = Pr{TH} = (1/2)(1/2) = 1/4
Pr{b-1(3)} = Pr{TTH} = (1/2)(1/2)(1/2) = 1/8
Pr{b-1(k)} = (1/2)k
E(b) = ?