02/01/2021
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The following topics will be covered in this lecture:
Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition
Multiplication Rule (for counting techniques)
Assume an operation can be described as a sequence of \( k \) steps, and
- \( n_1 \) is the number of ways to complete step 1
- \( n_2 \) is the number of ways to complete step 2 for each way to complete step 1, and
- \( n_3 \) is the number of ways to complete step 3 for each way to complete step 2, and
- so forth.
- Then the total number of ways to complete the operation is \[ n_1\times n_2\times \dots \times n_k \]
The number of permutations of \( n \) different elements is \( n! \) where \[ n! = n\times(n-1)\times(n-2)\times \dots \times 2 \times 1 \]
The number of permutations of subsets of \( r \) elements selected from a set of \( n \) different elements is \[ P_r^n = n\times(n-1)\times(n-2)\times\dots\times(n-r+1)=\frac{n!}{(n-r)!} \]
The number of permutations of \( n = n_1 + n_2 +\dots + n_r \) objects of which \( n_1 \) are of one type, \( n_2 \) are of a second type, … , and \( n_r \) are of an \( r \)-th type is \[ \frac{n!}{n_1!n_2!n_3!\dots n_r!} \]
The number of combinations, subsets of \( r \) elements that can be selected from a set of \( n \) elements, is denoted as \( C_r^n \) or \( \binom{n}{r} \) \[ C_r^n = \binom{n}{r}= \frac{n!}{r!(n-r)!} \]
Counting rules like the previous ones form the basis for computing simple probabilities.
The classical model for probability is based on a finite, discrete sample space where all simple events are equally likely.
We know that half of the cards are black and half of the cards are red.
\[ 26 \times 25 \]
given each choice, sampling without replacement.
We can also count all possible ways to draw any two cards:
\[ 52 \times 51. \]
Courtesy of Mario Triola, Essentials of Statistics, 6th edition
Equally Likely Outcomes:
Whenever a sample space consists of \( N \) possible outcomes that are equally likely, the probability of each outcome is \( 1/N \).
Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition
The probability of an event is determined as the ratio of the number of outcomes in the event to the number of outcomes in the sample space (for equally likely outcomes).
Probability of an event
For a discrete sample space, the probability of an event \( E \), denoted as \( P(E) \), equals the sum of the probabilities of the outcomes in \( E \).
Question: what is the probability that exactly 2 defective parts are selected in the sample?