STAT 230 - Chapter 1

• Classical Definition - $\frac{\text{number of ways the event can occur}}{\text{number of outcomes in S}}$
• Relative Frequency - Probability of an event is proportion of times the event occurs in a long series of repetitions of an experiment / process.
  • E.g probability of getting a 2 from a rolled dice is $\frac{1}{6}$
• Subjective Probability - Probability of an event is a measure of how sure person making the statement is that the event will happen
• Discrete - Consists of a finite or countably infinite set of sample points
• Simple definition
  • $0\le P(a_i)\le 1$
  • ${\sum }_{\text{all i}}P(a_i)={\sum }_{\text{all }{a}_i}P(a_i)=1$
  • Let $S=\{a_1,a_2,a_3,\dots \}$ be a discrete sample space
• Odds in favor of an event A is the probability the event occurs divided by the probability it does not occur, or simply $\frac{P(A)}{1-P(A)}$, odds against the event is reciprocal of this or simply $\frac{1-P(A)}{P(A)}$
• Additional Rule - Do job 1 in p ways and job 2 in q ways, then we can do either job 1 OR job 2 (but not both) in p + q ways
• Multiplication Rule Do job 1 in p ways and for each of these ways we can do job 2 in q ways. Then we can do both job 1 AND job 2 in p x q ways
• $n\times (n-1)\times \cdots \times 1$ ordered arrangements of length n using each symbol once and only once. This is denoted by $n!$
• $n\times (n-1)\times \cdots \times (n-k+1)$ ordered arrangements of length $k\le n$ each symbol at most once. This product is denoted by $n^{(k)}$ (read “n to k factors”). Note that $n^{(k)}=\frac{n!}{(n-k!)}$