Counting

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Published: July 06, 2021

This post covers Introduction to probability from Statistics for Engineers and Scientists by William Navidi.

Basic Ideas

Counting Methods
- The Fundamental Principle of Counting
  - If an operation can be performed in $n_{1}$ ways, and if for each of these ways a second operation can be performed in $n_{2}$ ways, then the total number of ways to perform the two operations is $n1*n2$.
  - Five lifeguards are available for duty one Saturday afternoon. There are three lifeguard stations. In how many ways can three lifeguards be chosen and ordered among the stations?
- Permutation
  - A permutation is an ordering of a collection of objects.
  - For example, there are six permutations of the letters $A, B, C$:
    - $ABC, ACB, BAC, BCA, CAB, and ~ CBA$.
  - For any positive integer $n$, $n! = n(n − 1)(n − 2)⋯(3)(2)(1)$.
  - Also, we define $0! = 1$.
  - The number of permutations of k objects chosen from a group of n objects is $\frac {n!} {(n − k)!}$
- Combinations
  - The number of combinations of k objects chosen from a group of n objects is:
    - $n \choose k$$ = {\frac {n!} {k!(n − k)!}}$
  - The number of ways of dividing a group of n objects into groups of $k_{1},\ldots, k_{r}$
    objects, where $k_{1} +\ldots+ k_{r} = n$, is:
    $ \frac {n!}{k_{1}!⋯k_{r}!}$.
A box of bolts contains $8$ thick bolts, $5$ medium bolts, and $3$ thin bolts. A box of nuts contains 6 that fit the thick bolts, $4$ that fit the medium bolts, and $2$ that fit the thin bolts. One bolt and one nut are chosen at random.What is the probability that the nut fits the bolt?