Study Notes for Chapter 5: The Binomial Probability Distribution
Understandable Statistics, 13th Edition
Chapter 5: The Binomial Probability Distribution and Related Topics
This chapter delves into the binomial distribution, its properties, visualizations, computations of mean and standard deviation, and practical examples.
5.3 Additional Properties of the Binomial Distribution
Discusses further complexities and applications of the binomial distribution.
Section Objectives
By the end of this section, you should be able to:
Make histograms for binomial distributions.
Compute the mean ($ $) and standard deviation ($ $) for a binomial distribution.
Determine through trial and error the minimum number of trials n needed to achieve a given probability of success P(r).
Binomial Distribution - Graphs
Importance of visualizing binomial distributions to understand their shape.
Graphing a Binomial Distribution
Key aspects of graphing binomial distributions:
Visualization helps comprehend the distribution shape.
The example used involves a waiter serving lone diners and their tipping behavior.
Example 1
Situation: A lone diner at a restaurant has a probability of 0.5 of leaving a tip. A waiter serves 6 diners.
Task: Create a graph of the binomial probability distribution showing probabilities for 0 to 6 diners giving tips.
Example 1: Solution (1 of 3)
Details of the Experiment:
This is a binomial experiment with 6 trials.
Success: A lone diner leaves a tip with:
Probability of success ($p$) = 0.5
Probability of failure ($q$) = 1 - $p$ = 0.5
Thus:
$n = 6$
$p = 0.5$
$q = 0.5$
Example 1: Solution (2 of 3)
Probability Distribution:
$r$ (number of successes) and corresponding $P(r)$ (probabilities):
$r = 0, P(r) = 0.016$
$r = 1, P(r) = 0.094$
$r = 2, P(r) = 0.234$
$r = 3, P(r) = 0.312$
$r = 4, P(r) = 0.234$
$r = 5, P(r) = 0.094$
$r = 6, P(r) = 0.016$
Example 1: Solution (3 of 3)
Graph Construction:
Place $r$ values on the horizontal axis.
Place $P(r)$ values on the vertical axis.
Construct a bar for each $r$ value extending from $r - 0.5$ to $r + 0.5$.
Height of each bar corresponds to $P(r)$.
Activity 1 – Using R
Graphing a binomial distribution using R:
Notice the symmetry.
Investigate the shape's skewness by varying p.
Code example provided:
plot(x, dbinom(x, n, p), type = 'h')Example:
plot(x, dbinom(x, 7, 0.2), type = 'h')
What Does the Graph of a Binomial Probability Distribution Tell Us?
Analysis of binomial distribution graphs provides insights:
Displays the probability of $r$ successes for each $r$ from 0 to $n$.
Mound-shaped and symmetric indicates $p$ close to 0.50.
Skewed right suggests $p < 0.50$; more right skewed means $p$ is closer to 0.
Skewed left indicates $p > 0.50$; more left skewed means $p$ is closer to 1.
Activity - Exercises 5.3 1–9, odd numbers
Practical application and reinforcement through exercises on the topic discussed.
Binomial Distribution – Mean and Standard Deviation
Key Mean and Standard Deviation Formulas:
For a random variable $X$ ~ B(n,p):
Mean: $ = n p$
Standard Deviation: $s = igr(
npqigr)^{1/2}$Note: Derivation of these results is not covered in the current course but may be explored in advanced modules.
Example 2
Consider a basketball player where:
Probability of successful free throw = 0.8
Assume shots taken independently.
Tasks:
(a) Expected number of successful shots out of 6.
(b) Standard deviation of the distribution.
Example 2: Solution
Let $X$ be the number of successful free throws. Thus:
$X ~ B(6, 0.8)$
Expected Value:
$ = np = (6)(0.80) = 4.8$
Standard Deviation:
$s = igr(
(6)(0.80)(0.20)igr)^{1/2}
= 0.98$
Activity
Satellite operation requires a minimum of 3 functioning solar cells:
Probability of failure for any cell = 0.15
Tasks:
(a) Determine the least number of cells needed to ensure expected power.
(b) Determine the least number of cells for a 97% chance of adequate power.