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?