In this investigation we will explore the development of the probability density function that defines the binomial random variable.
Coloured pens in a pencil case
Scenario one
Tom has five coloured pens in his pencil case, three blue and two green. Each time he starts class, he reaches into his pencil case and chooses one at random, without looking.
Our point of interest in this scenario will be how many times he selects a green pen in his first 2 classes on one day of school.
Task 1: To explore the probabilities and possibilities we can draw a probability tree diagram. Copy and complete the probability tree diagram below.
Task 2: We can define the outcomes of our "experiment" as the number of green pens selected by Tom in one day of five lessons. By looking at your probability tree diagram, list all possible outcomes.
Task 3: We will use the following table to tabulate the various combinations and their associated probabilities.
| Number of green pens | List of combinations | Probability calculations | Probability |
|---|---|---|---|
| 0 | BB | ||
| 1 | BG, GB | 0.48 | |
| 2 | 1\times0.4^2\times0.6^0 |
- Explain the probability calculations shown, and what each term of the calculation represents.
- Copy and complete the table.
Task 4: Explain why this scenario represents a discrete random variable.
Scenario two
Tom takes his pencil case with his two green pens and three blue pens to his first 3 classes of the day, again selecting one pen at random each lesson.
Task 1: Copy and complete the tree diagram below.
Task 2: Let X be the number of times Tom randomly chose a green pen that day. Copy and complete the table below.
| x | Number of combinations | Probability calculation | P(X=x) |
|---|---|---|---|
| 0 | |||
| 1 | 3 | ||
| 2 | 3\times0.4^2\times0.6^1 | 0.288 | |
| 3 | 1 |
Scenario three
Tom takes his pencil case with his two green pens and three blue pens to all his 5 classes of the day, again selecting one pen at random each lesson.
Task 1: Let X be the number of times Tom randomly chose a green pen that day. Copy and complete the table below.
| x | Number of combinations | Probability calculations | P(X=x) |
|---|---|---|---|
| 0 |
|
||
| 1 | 5\times0.4^1\times0.6^4 | 0.2592 | |
| 2 | 10 | ||
| 3 | 10\times0.4^3\times0.6^2 | 0.2304 | |
| 4 | 0.0768 | ||
| 5 |
1 |
Task 2: By referring to Pascal's triangle and/or combinations, describe how the column titled Number of combinations is obtained.
Task 3: Hence or otherwise expand and then calculate the value of each term from the expression (0.4+0.6)^5, and compare your result with your entries in the table.
Task 4: By examining the probability calculations and your results from Task 3, can you describe what extra feature you notice that makes this scenario a binomial discrete random variable, rather than a generic discrete random variable?
A second pencil case with coloured pens
Scenario four
Tanya also has a pencil case that she brings to each of her 5 classes each day and she also selects a pen at random each lesson. All we know is that the probability that she randomly selects a green pen each lesson has a value of p. The probability that she randomly selects any other coloured pen is q.
Task 1: The probability that Tanya selects only one green pen that day can be expressed as 5\times p^1\times q^4. Write an expression for the probability that Tanya selects only 3 green pens in a given day.
Task 2: Let X be the number of green pens Tanya selects in one school day. Copy and complete the following table, giving the probabilities in terms of p and q.
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| P(X=x) | 5\times p^1\times q^4 | 1\times p^5\times q^0 |
One final pencil case of coloured pens
Scenario five
Toby goes to a different school and attends n classes per day. He also selects a pen randomly from his pencil case each class, and the probability of Toby selecting a green pen is p.
Task 1: In terms of p, what is the probability that Toby does not select a green pen?
Task 2: The probability that Toby selects exactly two green pens in one day can be expressed as \binom{n}{2} p^2\left(1-p\right)^{n-2}. Explain the first term of this expression, \binom{n}{2}
Task 3: Write an expression for the probability that Toby selects exactly 4 green pens in a given day.
Task 4: Write an expression for the probability that Toby selects exactly k green pens in a given day.
To confirm your findings to this final task, proceed to the next chapter on the binomial distribution.