M1 Statistics: Applications of the Binomial and Poisson Distributions
Hey everyone! Welcome to one of the most practical topics in M1 Statistics. In the previous chapters, you learned about two powerful probability tools: the Binomial distribution and the Poisson distribution. Now, we're going to put them to work!
This chapter is all about becoming a probability detective. You'll learn how to look at a real-world problem and figure out which tool is the right one for the job. It's like knowing when to use a hammer and when to use a screwdriver. Master this, and you'll be able to solve a huge range of problems, from quality control in a factory to predicting customer arrivals at a shop. Let's get started!
1. Quick Recap: Meet the Distributions
Before we apply them, let's have a quick memory refresh of our two main players. Don't worry if you're a bit rusty, this will get you up to speed!
The Binomial Distribution: Counting Successes
Think of the Binomial distribution when you have a situation with a fixed number of trials, and each trial has only two possible outcomes (like success/failure, yes/no, defective/not defective).
Analogy: Imagine you're taking a 10-question multiple-choice quiz just by guessing. Each question is a 'trial'. You either get it right ('success') or wrong ('failure'). The Binomial distribution helps us find the probability of getting exactly, say, 3 questions right.
Conditions for Binomial Distribution (Remember B.I.N.S.!)
For a situation to be modelled by a Binomial distribution, it must satisfy four conditions:
• Binary: There are only two possible outcomes for each trial (success or failure).
• Independent: The outcome of one trial does not affect the outcome of another.
• Number of trials: The number of trials, n, is fixed in advance.
• Same probability: The probability of success, p, is the same for each trial.
The Formulas You Need
If a random variable X follows a binomial distribution, we write $$X \sim B(n, p)$$
• Probability Formula: The probability of getting exactly k successes in n trials is:
$$ P(X=k) = C_k^n p^k (1-p)^{n-k} $$• Mean (Expected Value): The average number of successes.
$$ E(X) = np $$• Variance: A measure of how spread out the results are.
$$ Var(X) = np(1-p) $$Key Takeaway
Use the Binomial distribution when you're counting the number of 'successes' in a fixed number of attempts (n).
The Poisson Distribution: Counting Events in an Interval
Think of the Poisson distribution when you are counting the number of times an event occurs over a fixed interval of time, area, or space. The key is that these events happen randomly and at a constant average rate.
Analogy: Imagine you're working at a call centre. You want to know the probability of receiving exactly 5 calls in the next hour. You know that on average, you get 8 calls per hour. The Poisson distribution is the perfect tool for this.
Conditions for Poisson Distribution
• Events occur at a constant average rate (denoted by λ).
• Events are random and independent of each other (one call arriving doesn't make another one more or less likely to arrive).
The Formulas You Need
If a random variable X follows a Poisson distribution, we write $$X \sim Po(\lambda)$$ where λ (lambda) is the average number of events in the given interval.
• Probability Formula: The probability of exactly k events occurring in the interval is:
$$ P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} $$• Mean (Expected Value): This is just the average rate!
$$ E(X) = \lambda $$• Variance: Super easy to remember!
$$ Var(X) = \lambda $$Did you know?
For the Poisson distribution, the mean and the variance are always the same! This is a unique property and can sometimes be a hint in exam questions.
Key Takeaway
Use the Poisson distribution when you're counting events over a continuous interval (like time or space) and you're given an average rate (λ).
2. The Main Event: How to Choose the Right Distribution
This is the most important skill in this chapter. When you read a problem, you need to look for clues to decide whether it's a Binomial or a Poisson problem. Here’s a handy comparison table to help you decide.
| Feature | Binomial Distribution | Poisson Distribution | | ----------------------- | ----------------------------------------------------------- | --------------------------------------------------------- | | What are we counting? | Number of successes | Number of events | | The setup? | In a fixed number of trials (n) | In a fixed interval (of time, space, etc.) | | Key Parameters | n (number of trials) and p (probability of success) | λ (average number of events in the interval) | | Clue Words | "Out of 20 items...", "15 coin flips...", "sample of 50..." | "Per hour...", "per square metre...", "in a minute..." | | Example | Probability of 3 heads in 10 coin flips. | Probability of 5 typos on a single page of a book. |A Simple Decision Guide
When you read a problem, ask yourself these questions:
1. Am I given a fixed number of attempts (like n=20) and a probability of success for each attempt (like p=0.1)?
If yes, it’s almost certainly Binomial.
2. Am I given an average rate of something happening over a period of time or space (like 3 customers per hour)?
If yes, it’s almost certainly Poisson.
Don't worry if this seems tricky at first. The more problems you practice, the faster you'll become at spotting the clues!
3. Worked Examples: Putting Theory into Practice
Example 1: The Basketball Player (Binomial)
A basketball player has a 70% chance of making a free throw. If she takes 8 free throws in a game, what is the probability that she makes exactly 6 of them?
Step-by-Step Solution:
1. Identify the Distribution:
• Are there a fixed number of trials? Yes, n = 8 free throws.
• Are there two outcomes? Yes, make ('success') or miss ('failure').
• Is the probability of success constant? Yes, p = 0.7 for each throw.
• Are the trials independent? Yes, one throw doesn't affect the next.
This fits the B.I.N.S. conditions perfectly. So, we use the Binomial distribution! We have $$X \sim B(8, 0.7)$$.
2. Identify the Variables:
• n = 8
• p = 0.7
• 1-p = 0.3
• k = 6 (we want exactly 6 successes)
3. Apply the Formula:
$$ P(X=6) = C_6^8 (0.7)^6 (0.3)^{8-6} $$
$$ P(X=6) = 28 \times (0.7)^6 \times (0.3)^2 $$
$$ P(X=6) = 28 \times 0.117649 \times 0.09 $$
$$ P(X=6) \approx 0.2965 $$
4. Final Answer:
The probability of her making exactly 6 free throws is about 0.2965 (or 29.65%).
Example 2: The Coffee Shop (Poisson)
A small coffee shop gets an average of 4 customers per 10-minute period. What is the probability that exactly 3 customers arrive in a 10-minute period? What about in a 5-minute period?
Part A: Probability in a 10-minute period
1. Identify the Distribution:
We are counting events (customer arrivals) over a fixed interval (10 minutes) and we are given an average rate. This is a classic Poisson problem!
2. Identify the Variables:
• The interval is 10 minutes.
• The average rate for this interval is λ = 4.
• We want to find the probability of exactly k = 3 events.
So, $$X \sim Po(4)$$.
3. Apply the Formula:
$$ P(X=3) = \frac{e^{-4} 4^3}{3!} $$
$$ P(X=3) = \frac{e^{-4} \times 64}{6} $$
$$ P(X=3) \approx 0.018315 \times \frac{64}{6} \approx 0.1954 $$
4. Final Answer:
The probability of exactly 3 customers arriving in 10 minutes is about 0.1954.
Part B: Probability in a 5-minute period
1. Adjust the λ! (This is a common trick!)
The question is now about a 5-minute interval, not a 10-minute one. We must adjust our average rate λ to match the new interval.
• Original rate: 4 customers per 10 minutes.
• New rate: The interval is half as long, so the average number of customers will also be half.
• New λ = 4 × (5 / 10) = 2 customers per 5 minutes.
So, for this part, we use $$Y \sim Po(2)$$.
2. Apply the Formula with the new λ:
We still want to find the probability of exactly 3 customers, so k = 3.
$$ P(Y=3) = \frac{e^{-2} 2^3}{3!} $$
$$ P(Y=3) = \frac{e^{-2} \times 8}{6} $$
$$ P(Y=3) \approx 0.1353 \times \frac{8}{6} \approx 0.1804 $$
Common Mistakes to Avoid
• Forgetting to adjust λ! Always check if the time interval in the question is the same as the interval given for the average rate. If not, you MUST adjust λ proportionally.
4. The Special Case: Poisson Approximation to Binomial
Sometimes, a problem looks Binomial, but the numbers are horrible to calculate. Imagine $$X \sim B(2000, 0.001)$$. Calculating $$C_{2}^{2000}$$ is a nightmare!
Luckily, there's a shortcut. When n is very large and p is very small, the Binomial distribution starts to look a lot like the Poisson distribution. We can use Poisson as an easy approximation!
When can we use this approximation?
Use it when, in a Binomial distribution:
• n is large (a good rule of thumb is n > 50)
• p is small (a good rule of thumb is p < 0.1)
How to do it?
It's simple! You just set the Poisson average rate λ equal to the Binomial mean (np).
The magic step: Calculate $$ \lambda = np $$
Example 3: The Defective Chips (Poisson Approximation)
A factory produces computer chips, and 0.2% of them are defective. In a random sample of 1000 chips, what is the probability that exactly 4 are defective?
Step-by-Step Solution:
1. Identify the Original Distribution:
This is a Binomial problem. We have a fixed number of trials (n=1000) and a constant probability of success (a chip being defective, p=0.002). So, $$X \sim B(1000, 0.002)$$.
2. Check if Approximation is Suitable:
• n = 1000 (This is very large! ✓)
• p = 0.002 (This is very small! ✓)
The conditions are perfect. We can use the Poisson approximation. It will be much easier than calculating $$C_4^{1000}$$!
3. Calculate the new λ:
$$ \lambda = np = 1000 \times 0.002 = 2 $$
4. Use the Poisson Formula:
We now pretend this is a Poisson problem with λ=2, and we want to find the probability of k=4.
So we use $$X' \sim Po(2)$$.
$$ P(X'=4) = \frac{e^{-2} 2^4}{4!} $$
$$ P(X'=4) = \frac{e^{-2} \times 16}{24} $$
$$ P(X'=4) \approx 0.1353 \times \frac{16}{24} \approx 0.0902 $$
5. Final Answer:
The approximate probability of finding exactly 4 defective chips is about 0.0902.
Key Takeaway
If you see a Binomial problem with a huge n and a tiny p, your brain should scream: "Poisson Approximation!" Just calculate λ=np and solve it as a simple Poisson problem.