Title: Binomial and Geometric Distributions
1Binomial and Geometric Distributions
2Properties of Binomial Experiments
- It consists of a fixed number of observations
called trials. - Each trial can result in one of only two mutually
exclusive outcomes success (S) and failure (F) - Outcomes of different trials are independent.
- The probability that a trial results in S is the
same for each trial.
3Binomial Random Variable
- The binomial random variable x is defined as
- x number of successes observed when an
experiment is performed - The probability distribution of x is called the
binomial probability distribution.
4Example
- A few days ago we studied the purchasing
characteristics of customers shopping for a hot
tub. We considered x number among four
customers who selected an electric (as opposed to
gas) hot tub. - This is a binomial experiment with
- number of trials 4 P(success) P(E)0.4
5New Example
- Lets look at the case of five customers, a
binominal experiment with five trials. - Possible x values are 0, 1, 2, 3, 4, 5.
- How many possible outcomes are there? (Remember
the fundamental counting principle? - There are 32 possible outcomes, and five of them
yield x1. - SFFFF FSFFF FFSFF FFFSF FFFFS
6Example (cont)
- Lets take a look at the probability for the
outcomes x1 (SFFFF, FSFFF, FFSFF, ) - The calculation will be the same for any outcome
with only one success (x1).
7Example (cont)
8Example (cont)
- What about for x2. How many ways can we select
two from among the five trials to be the Ss? - What is the probability of P(SSFFF)?
9Example (cont)
- What is p(2) if we know
- Remember
10The Binomial Distribution
- Ready for this
- Let
- n number of independent trials in a binomial
experiment - p constant probability that any particular
trial results in a success
11Example (Revisited)
- Lets go back to our 5 customers and look at p(2)
using our new formula. - We said.
- Does that fit with our formula?
- n 5 x2 p.4
12Next Example
- 60 of all watches sold by a large discount store
have a digital display and the rest have analog.
The type of watch purchased by the next 12
customers will be noted. - x number of watches that have a digital
display.
13Next Example
- What is the probability that 4 watches are
digital? Go ahead and try it now
14Using Appendix Table X
- To find p(x) for any particular value of x,
- Locate the part of the table corresponding to
your value of n (5, 10, 15, 20, or 25) - Move down to the row labeled with your value of
x. - Go across to the column headed by the specified
value of p - Try it now lets do n20 and p 0.8
- Find p(15).
15Homework
- 7.42
- 7.43 (cant use table)
- 7.44 (cant use table)
- 7.45 (use table)
- 7.47
- 7.48
- 7.49
- 7.50