Binomial and Poisson Probability Distributions for SS 3

Class: SSS 3
Term: First Term
Week: 9
Age: 16-17 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Probability and Statistics
Previous Lesson: Correlation and Rank Correlation
Topic: PROBABILITY DISTRIBUTION AND APPROXIMATIONS
Subject Matter: Binomial; Poisson; Binomial approximation by Poisson; Normal distribution; Normal approximation by binomial distribution

Specific Objectives

By the end of the lesson, pupils/students should be able to:

Cognitive Domain

• Define Binomial, Poisson, and Normal probability distributions.
• State the conditions for applying each distribution.
• Calculate probabilities using the Binomial, Poisson, and Normal distribution formulas.
• Explain the conditions for approximating Binomial by Poisson.
• Explain the conditions for approximating Binomial by Normal.
• Solve problems involving these distributions and their approximations.

Affective Domain

• Appreciate the importance of probability distributions in solving real-world problems.
• Develop a systematic approach to solving probability distribution problems.

Psychomotor Domain

• Apply appropriate formulas to solve problems related to probability distributions.
• Demonstrate the use of continuity correction when approximating Binomial by Normal.

Reference Materials

The following resources were used in planning this lesson:

• 2025 Revised 9 Years Basic Education Curriculum
• Relevant State Unified Scheme of Work
• Further Mathematics for Senior Secondary Schools (Textbook)
• The HeadTeacher Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• Whiteboard and markers
• Scientific calculator
• Further Mathematics textbook
• Normal distribution tables (Z-tables)

Rationale for the Lesson

This lesson is essential for students to understand how to model and analyse random phenomena using different probability distributions. It provides them with tools to calculate probabilities for various scenarios and to approximate complex distributions with simpler ones under specific conditions. This knowledge is fundamental for advanced studies in statistics, engineering, and data science.

Prerequisite/Previous Knowledge

Students should have a basic understanding of probability, permutations, combinations, and basic statistical concepts like mean and variance.

Lesson Content/Board Summary

PROBABILITY DISTRIBUTION AND APPROXIMATIONS

1. Binomial Distribution

The Binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant for each trial.

Conditions for Binomial Distribution:

1. The number of trials, (n), is fixed.
2. Each trial has only two possible outcomes: success or failure.
3. The probability of success, (p), is constant for each trial.
4. The trials are independent.

Formula:

(P(X=k) = binom{n}{k} p^k (1-p)^{n-k})

Where:

1. (P(X=k)) = Probability of (k) successes in (n) trials
2. (n) = Number of trials
3. (k) = Number of successes
4. (p) = Probability of success on a single trial
5. ((1-p)) or (q) = Probability of failure on a single trial
6. (binom{n}{k} = frac{n!}{k!(n-k)!}) = Binomial coefficient

Mean and Variance:

1. Mean (E(X) = np)
2. Variance (Var(X) = np(1-p))

Example 1

Question: A fair coin is tossed 10 times. What is the probability of getting exactly 7 heads?

Solution:

Step 1: Identify parameters.

(n = 10) (number of trials)
(k = 7) (number of successes)
(p = 0.5) (probability of getting a head)
(q = 1 – p = 0.5) (probability of getting a tail)

Step 2: Write the Binomial formula.

(P(X=k) = binom{n}{k} p^k q^{n-k})

Step 3: Substitute the values.

(P(X=7) = binom{10}{7} (0.5)^7 (0.5)^{10-7})
(P(X=7) = frac{10!}{7!3!} (0.5)^7 (0.5)^3)
(P(X=7) = frac{10 times 9 times 8}{3 times 2 times 1} (0.5)^{10})
(P(X=7) = 120 times 0.0009765625)

Step 4: Simplify and write the answer.

(P(X=7) = 0.1171875)

Answer: The probability of getting exactly 7 heads is (0.1172) (to 4 decimal places).

2. Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event.

Conditions for Poisson Distribution:

1. Events occur independently.
2. Events occur at a constant average rate ((lambda)).
3. The probability of an event occurring in a short interval is proportional to the length of the interval.
4. The probability of more than one event occurring in a very short interval is negligible.

Formula:

(P(X=k) = frac{e^{-lambda} lambda^k}{k!})

Where:

1. (P(X=k)) = Probability of (k) events occurring
2. (lambda) (lambda) = Average rate of events in the given interval (mean number of occurrences)
3. (e) = Euler’s number (approximately 2.71828)
4. (k) = Number of events
5. (k!) = Factorial of (k)

Mean and Variance:

1. Mean (E(X) = lambda)
2. Variance (Var(X) = lambda)

Example 1

Question: The average number of calls received by a call centre in an hour is 5. What is the probability that exactly 3 calls are received in an hour?

Solution:

Step 1: Identify parameters.

(lambda = 5) (average number of calls per hour)
(k = 3) (number of calls to occur)

Step 2: Write the Poisson formula.

(P(X=k) = frac{e^{-lambda} lambda^k}{k!})

Step 3: Substitute the values.

(P(X=3) = frac{e^{-5} 5^3}{3!})
(P(X=3) = frac{0.006738 times 125}{6})
(P(X=3) = frac{0.84225}{6})

Step 4: Simplify and write the answer.

(P(X=3) = 0.140375)

Answer: The probability of receiving exactly 3 calls in an hour is (0.1404) (to 4 decimal places).

3. Binomial Approximation by Poisson Distribution

The Poisson distribution can be used to approximate the Binomial distribution when the number of trials (n) is large and the probability of success (p) is small.

Conditions for Approximation:

1. (n ge 50) (number of trials is large)
2. (p le 0.1) (probability of success is small)

When these conditions are met, the mean of the Poisson distribution, (lambda), is calculated as (lambda = np).

Example 1

Question: In a manufacturing process, 1% of items are defective. If a random sample of 200 items is taken, use the Poisson approximation to find the probability that exactly 3 items are defective.

Solution:

Step 1: Identify Binomial parameters and check approximation conditions.

(n = 200)
(p = 0.01)
Since (n = 200 ge 50) and (p = 0.01 le 0.1), Poisson approximation is appropriate.

Step 2: Calculate (lambda) for the Poisson distribution.

(lambda = np = 200 times 0.01 = 2)

Step 3: Use the Poisson formula with (lambda = 2) and (k = 3).

(P(X=3) = frac{e^{-2} 2^3}{3!})
(P(X=3) = frac{0.135335 times 8}{6})
(P(X=3) = frac{1.08268}{6})

Step 4: Simplify and write the answer.

(P(X=3) = 0.180447)

Answer: The probability that exactly 3 items are defective is approximately (0.1804) (to 4 decimal places).

4. Normal Distribution

The Normal distribution is a continuous probability distribution that is symmetrical about its mean, forming a bell-shaped curve. It is widely used to model many natural phenomena.

Characteristics:

1. It is bell-shaped and symmetrical about its mean.
2. The mean, median, and mode are all equal and located at the centre of the distribution.
3. The total area under the curve is 1.
4. It extends infinitely in both directions, approaching the horizontal axis but never touching it.

Parameters:

1. Mean ((mu)): Determines the centre of the distribution.
2. Standard Deviation ((sigma)): Determines the spread of the distribution.

Standard Normal Distribution:

A special case of the Normal distribution with a mean of 0 and a standard deviation of 1. Any Normal distribution can be converted to a standard normal distribution using the Z-score formula.

Z-score Formula:

(Z = frac{X – mu}{sigma})

Where:

1. (Z) = Z-score (number of standard deviations (X) is from the mean)
2. (X) = Value from the original distribution
3. (mu) = Mean of the original distribution
4. (sigma) = Standard deviation of the original distribution

Example 1

Question: The scores on a test are normally distributed with a mean of 60 and a standard deviation of 10. What is the Z-score for a student who scored 75?

Solution:

Step 1: Identify parameters.

(X = 75)
(mu = 60)
(sigma = 10)

Step 2: Write the Z-score formula.

(Z = frac{X – mu}{sigma})

Step 3: Substitute the values.

(Z = frac{75 – 60}{10})
(Z = frac{15}{10})

Step 4: Simplify and write the answer.

(Z = 1.5)

Answer: The Z-score for a student who scored 75 is 1.5.

5. Normal Approximation by Binomial Distribution

The Normal distribution can be used to approximate the Binomial distribution when the number of trials (n) is large enough.

Conditions for Approximation:

1. (n ge 30) (number of trials is large)
2. (np ge 5)
3. (n(1-p) ge 5)

When these conditions are met, the Binomial distribution can be approximated by a Normal distribution with:

1. Mean (mu = np)
2. Standard Deviation (sigma = sqrt{np(1-p)})

Continuity Correction:

Since the Binomial distribution is discrete and the Normal distribution is continuous, a continuity correction factor of 0.5 must be applied when using the Normal approximation. This adjusts the discrete values to a continuous range.

1. For (P(X=k)), use (P(k-0.5 < X < k+0.5)).
2. For (P(X ge k)), use (P(X > k-0.5)).
3. For (P(X le k)), use (P(X < k+0.5)).
4. For (P(X > k)), use (P(X > k+0.5)).
5. For (P(X < k)), use (P(X < k-0.5)).

Example 1

Question: A fair coin is tossed 100 times. Use the Normal approximation to find the probability of getting exactly 55 heads.

Solution:

Step 1: Identify Binomial parameters and check approximation conditions.

(n = 100)
(p = 0.5)
(np = 100 times 0.5 = 50)
(n(1-p) = 100 times 0.5 = 50)
Since (n = 100 ge 30), (np = 50 ge 5), and (n(1-p) = 50 ge 5), Normal approximation is appropriate.

Step 2: Calculate mean and standard deviation for the Normal approximation.

(mu = np = 50)
(sigma = sqrt{np(1-p)} = sqrt{100 times 0.5 times 0.5} = sqrt{25} = 5)

Step 3: Apply continuity correction for (P(X=55)).

This corresponds to (P(54.5 < X < 55.5)) in the Normal distribution.

Step 4: Calculate Z-scores for (X_1 = 54.5) and (X_2 = 55.5).

(Z_1 = frac{54.5 – 50}{5} = frac{4.5}{5} = 0.9)
(Z_2 = frac{55.5 – 50}{5} = frac{5.5}{5} = 1.1)

Step 5: Use Z-tables to find probabilities.

(P(Z < 1.1) approx 0.8643)
(P(Z < 0.9) approx 0.8159)

Step 6: Calculate the desired probability.

(P(0.9 < Z < 1.1) = P(Z < 1.1) – P(Z < 0.9))
(P(0.9 < Z < 1.1) = 0.8643 – 0.8159 = 0.0484)

Answer: The probability of getting exactly 55 heads is approximately (0.0484) (to 4 decimal places).

Teaching Methods/Instructional Techniques

Explanation, Discussion, Question and Answer, Guided Practice, Problem Solving

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Review/Engage

Teacher’s Activity: The teacher greets the students and reviews basic concepts of probability, asking questions like, “What is probability?” and “What are some basic rules of probability?” The teacher then introduces the topic of probability distributions as a way to model random events.

Pupils’ Activity: Students respond to questions and actively participate in the review of basic probability concepts.

Learning Point: Recall of probability basics

Step 2: Binomial Distribution

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains the Binomial distribution, its conditions, formula, mean, and variance. The teacher presents a worked example on the board, guiding students through the calculation of Binomial probabilities.

Pupils’ Activity: Students listen, ask questions for clarification, and copy the definition, formula, and example into their notebooks.

Learning Point: Understanding Binomial distribution

Step 3: Poisson Distribution

Time: 8 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher introduces the Poisson distribution, explaining its conditions, formula, mean, and variance. The teacher provides a worked example, highlighting scenarios where Poisson distribution is applicable.

Pupils’ Activity: Students pay attention, note down the key points, and follow the steps in the example problem.

Learning Point: Understanding Poisson distribution

Step 4: Binomial Approximation by Poisson

Time: 6 minutes

Teaching Skill: Explanation/Guided Practice

Teacher’s Activity: The teacher explains when and why the Binomial distribution can be approximated by the Poisson distribution, stating the conditions ((n ge 50, p le 0.1)) and how to determine (lambda). The teacher guides students through a problem involving this approximation.

Pupils’ Activity: Students identify the conditions for approximation and solve the problem with teacher guidance.

Learning Point: Poisson approximation of Binomial

Step 5: Normal Distribution

Time: 6 minutes

Teaching Skill: Explanation/Illustration

Teacher’s Activity: The teacher explains the characteristics of the Normal distribution, its bell shape, symmetry, and parameters ((mu, sigma)). The teacher also introduces the concept of the Standard Normal Distribution and the Z-score formula, providing a simple example.

Pupils’ Activity: Students observe the illustration of the Normal curve, understand its properties, and learn the Z-score formula.

Learning Point: Characteristics of Normal distribution

Step 6: Normal Approximation by Binomial Distribution

Time: 6 minutes

Teaching Skill: Explanation/Application

Teacher’s Activity: The teacher explains the conditions for approximating the Binomial distribution by the Normal distribution ((n ge 30, np ge 5, n(1-p) ge 5)). The teacher emphasises the importance of continuity correction and demonstrates its application in a problem.

Pupils’ Activity: Students identify the conditions, understand continuity correction, and attempt to apply it in a given problem.

Learning Point: Normal approximation of Binomial

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. State two conditions for a Binomial distribution.
2. When can a Poisson distribution approximate a Binomial distribution?
3. What are the characteristics of a Normal distribution?
4. Explain the purpose of continuity correction when approximating Binomial by Normal.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Assessment of understanding

Step 8: Note-Taking

Time: 4 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes, including definitions, formulas, conditions, and examples, into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording lesson content

Step 9: Conclusion

Time: 2 minutes

Teaching Skill: Summarising

Teacher’s Activity: The teacher briefly summarises the main points of the lesson, reiterating the importance of understanding the different probability distributions and their approximations in real-world applications. The teacher encourages students to practice more problems.

Pupils’ Activity: Students listen attentively and ask any final questions.

Learning Point: Consolidation of key concepts

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebook:

1. A biased coin has a probability of 0.6 of landing on heads. If it is tossed 8 times, calculate the probability of getting exactly 5 heads.
2. The number of accidents on a particular road follows a Poisson distribution with an average of 2 accidents per week. Find the probability that there will be exactly 4 accidents in a given week.
3. A factory produces light bulbs, and 0.5% of them are defective. In a batch of 500 bulbs, use the Poisson approximation to find the probability that there are exactly 2 defective bulbs.
4. The heights of students in a school are normally distributed with a mean of 160 cm and a standard deviation of 8 cm. What is the Z-score for a student whose height is 172 cm?
5. In a survey, 70% of people prefer brand A. If 120 people are randomly selected, use the Normal approximation with continuity correction to find the probability that between 80 and 85 people (inclusive) prefer brand A.

Lesson Keywords

• Binomial Distribution – Probability distribution for a fixed number of independent trials with two outcomes.
• Poisson Distribution – Probability distribution for the number of events in a fixed interval.
• Normal Distribution – A continuous, bell-shaped, symmetrical probability distribution.
• Approximation – Using one distribution to estimate probabilities for another.
• Continuity Correction – Adjustment made when approximating a discrete distribution with a continuous one.
• Z-score – A measure of how many standard deviations an element is from the mean.
• Lambda ((lambda)) – The mean rate of events in a Poisson distribution.

Differentiation

For struggling learners, provide extra guided practice with simpler problems, focusing on identifying the correct distribution and formula. For advanced learners, challenge them with more complex problems involving cumulative probabilities or inverse problems using probability tables.

Suggested Lesson Videos

Search on YouTube for: “Binomial Poisson Normal distribution SS3 Further Maths”

Teacher Guide for Using This Lesson Plan

Before the lesson, ensure you have a scientific calculator and, if possible, a Normal distribution (Z-table) handy for demonstration during the Normal approximation section. Begin by clearly differentiating between discrete and continuous probability distributions. When explaining each distribution, present its real-world application to make the concept more concrete. For approximations, emphasise the conditions under which they are valid and walk students through the step-by-step process, especially for continuity correction in Normal approximation of Binomial. Encourage students to ask questions and work through examples on their own. Students should copy the Board Summary notes after the main teaching points have been covered and evaluated, typically in Step 8. Pay attention to common errors such as incorrect formula application or forgetting continuity correction. Provide individual support to slower learners and assign challenging problems to faster learners.