Download Bernoulli, binomial, and geometric

Student Worksheet: “How Lucky Are You?”
Class: _____
Level:
Name: _______________ (
)
Form 6 or 7 (AS level)
Domain: Introduction to three typical discrete probability distributions
(Bernoulli, binomial, and geometric)
Objectives:
1. To learn the definitions of Bernoulli, binomial, and geometric distributions
2. To know the basic properties (shape, mean, and variance) of the three discrete distributions
3.
To understand the inter-relationship among the three distributions
Prerequisite Knowledge:
1. Classification and presentation of data
2.
Basic definition and properties of discrete probability distributions
Preparation:
Try to use the simulation program a few times for the color balls drawing
process (provided by StatNet).
Activity:
1.
2
Think about the coin-tossing experiments (and have a try yourself):
- a coin is tossed once and observe whether the outcome is “head” or “tail”
- a coin is tossed 5 times and observe the number of “heads” obtained
- to count the number of tosses until the first “head” is obtained
Notice the following properties:
- each trial has only two outcomes: success (says “head”) or failure (says “tail”)
- the repeated trials are independent
- the probability of a success (p) remains constant from trial to trial
[p = 1/2 for an unbiased or balanced coin]
[Bernoulli]
[Binomial]
[Geometric]
Main Task:
1. Follow the procedures below:
1.1 Randomly pick 5 color balls from the computer simulation program provided by StatNet.
(This is identical to randomly draw 5 color balls with replacement from an oblique bag
that contains 4 orange and 6 white balls).
1.2 Record the exact color sequence of the 5 balls: O – orange, W - white
e.g. W1W2O3W4O5
(If you are doing this activity in your home, you can repeat the experiment for thirty times
to obtain the data set required.)
2. Use the following definitions of random variables to observe the values of C, X and T:
C represents the color of the first ball drawn: c = 0 (for white) or 1 (for orange)
X represents the number of orange balls obtained: x = 0, 1, 2, 3, 4, 5
T represents the number of tosses until the first orange ball is obtained: t = 1, 2, 3, 4, 5
(if all 5 balls are white, then t = 6+)
(For the example shown in step 1.2 above: C=0 , X=2, T=3)
3. Repeat step 1 and 2 for at least thirty times. You should make 10 draws first and then another
10 and so on in order to visualize the formulation of the final distribution. (Or you can use the
data set obtained all by yourself.)
4. Complete the frequency column of the following frequency table for X:
No. of orange balls, x
Frequency, f
Relative frequency
0
1
2
3
4
5
Total
5.
Construct the corresponding bar chart using a spreadsheet program such as Excel (optional).
6.
Complete the relative frequency column also [ probability distribution for X]
7.
Compute the mean and variance for X (which is called a binomial random variable):
mean of X = _____
variance of X = _____
Exercise:
Repeat tasks 4 to 7 for variables C (which is called a Bernoulli random variable);
and then complete the table below:
Color of the
first ball, c
0 (white)
1 (orange)
Total
Frequency, f
Relative
frequency
Mean of C =
Variance of C =
Repeat tasks 4 to 7 for variables T (which is called a geometric random variable);
and then complete the table below:
No. of draws to obtain
the first orange ball, t
1
2
3
4
5
6+
Total
Frequency, f
Relative frequency
Mean of T =
Variance of T =
Enrichment:
1.
Can you find out any analogies in the following situations?
- a sample survey conducted to predict voter preference for a particular candidate.
- a new company wants to know the fraction of home that use Internet for shopping.
- a teacher is interested in the percentage of students who pass the course.
- a new drug is effective or ineffective when administered to a patient.
- a manufactured item selected from a production line is defective or non-defective.
- with each contact, either a salesperson will consummate a sale or no sale will result.
2.
Challenging Tasks
- Try to explain why those are the results for C, X, and T. [Hint: p = 0.4]
- Derive the probability functions for the three variables. ( Please check with your textbook.)
- Consider the case of sampling without replacement
[Hypergeometric distribution]