Basic Probability

Probability theory provides a formal framework for reasoning about the likelihood of events.

Definitions

• An experiment is a procedure that yields one of a set of possible outcomes. As our ongoing example, consider the experiment of tossing two six-sided dice, one red and one blue, with each face bearing a distinct integer $\{1,\dots ,6\}$.
• A sample space $S$ is the set of possible outcomes of an experiment. In our dice example, there are 36 possible outcomes

$$\begin{array}{r}S=\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),\\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),\\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6),\\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}\end{array}$$

• An event $E$ is a specified subset of the outcomes of an experiment. The event that the sum of the dice equals 7 or 11 (the conditions to win at craps on the first roll) is the subset

$$E=\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1),(5,6),(6,5)\}$$

• The probability of an outcome $s$, denoted $p(s)$, is a number with the two properties:
  • For each outcome $s$ in sample space $S$, we have $0\le p(s)\le 1$
  • The sum of probabilities of all outcomes adds to one: ${\sum }_{s\in S}p(s)=1$
• In our example, if we assume two distinct fair dice, the probability $p(s)=(1/6)\times (1/6)=1/36$ for all outcomes $s\in S$.
• The probability of an event $E$ is the sum of probabilities of the outcomes of the event (members of event subset). Thus,

$$P(E)=\sum _{s\in E}p(s)$$

• The complement of the event $\bar{E}$, the case when $E$ does not occur. Then

$$P(E)=1-P(\bar{E})$$

• A random variable $V$ is s a numerical function on the outcomes of a probability space. The function “sum the values of two dice” ($V((a,b))=a+b$) produces an integer results between 2 and 12. This implies a probability distribution of the possible values of the random variable. The probability $P(V(s)=7)=1/6$, while $P(V(s)=12)=1/36$.
• The expected value of a random variable $V$ defined on a sample space $S$, $E(V)$, is defined

$$E(V)=\sum _{s\in S}p(s)\cdot V(s)$$