
# 10. Bernoulli Processes


The following topics are included in the seven videos below.

1. Intro to Bernoulli Processes
2. Bernoulli Processes, Example 1 and definitions
3. Bernoulli Processes, Example 2
4. Bernoulli Processes, Example 3
5. Bernoulli Processes, Example 4 (with conditional)
6. Bernoulli Processes, Example 5 (using a complement)
7. Bernoulli Processes, Example 6 (with conditional)

#### Prework

1. A fair die is tossed five times. Find the probability of throwing four sixes.

2. A fair die is tossed five times. Find the probability of throwing at least 1 six.

3. The patrons at a certain library check out only fiction books 70% of the time. Assume that the visits to the library are independent. What is the probability that 20 of the next 25 patrons will check out only fiction books?

1. A success is throwing a six while a failure is throwing anything else. On one toss, the probability of throwing a six is $$\frac{1}{6}$$ so $$p=\frac{1}{6}$$ and therefore $$1-p=\frac{5}{6}$$. We roll the die $$5$$ times so $$n=5$$ and we want four sixes, so $$r=4$$. Therefore, Pr(four sixes)$$=C(5,4)\cdot (\frac{1}{6})^4\cdot (\frac{5}{6})^1$$.
2. Everything is the same as in problem 1, except that $$r=1,2,3,4,5$$. We could plug each of these into the formula and add them together, or we we can use the complement. In the complement, $$r=0$$. Therefore Pr(at least 1 six)$$=1-C(5,0)\cdot (\frac{1}{6})^0\cdot (\frac{5}{6})^5$$.
3. A success is a library patron checking out only fiction while a failure is a library patron not checking out only fiction. We also see that $$p=0.7, 1-p=0.3, n=25$$ and $$r=20$$. So Pr(20 check out only fiction)$$=C(25,20)\cdot (0.7)^{20}\cdot (0.3)^5$$.