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Applications and Technology

Apply discrete distributions to real problems and learn to use technology

Lesson Objectives

By the end of this lesson, you will be able to:

Compare different discrete distributions and choose the right one
Apply binomial distribution to complex real-world scenarios
Use technology (calculators, software) to compute binomial probabilities
Understand when normal approximation can be used for binomial
Recognize other discrete distributions (Poisson preview)

1. Choosing the Right Distribution

Decision Flowchart:

Is the random variable discrete or continuous?
- Discrete → Continue to #2
- Continuous → Use continuous distributions (Module 5)
Does it satisfy the four binomial conditions?
- Yes → Use binomial distribution
- No → Determine which condition is violated
If not binomial, what's violated?
- Sampling without replacement → Hypergeometric
- Variable number of trials → Geometric or Negative Binomial
- Counting rare events in interval → Poisson

Example 1: Choosing Distributions

Scenario A: A company receives 500 customer calls per day. Count the number of complaint calls (historically 3% are complaints).

Analysis:

Discrete (counting)
Fixed n = 500
Two outcomes (complaint or not)
Independent (reasonable assumption)
Constant p = 0.03
Use BINOMIAL with n = 500, p = 0.03

Scenario B: Count the number of emails received in an hour (average is 15 per hour).

Analysis:

Discrete (counting)
No fixed n (not a set number of "trials")
This is counting events in a time interval
Use POISSON distribution (not binomial)

Scenario C: Deal cards until you get an ace. Count number of cards dealt.

Analysis:

Discrete (counting)
Variable n (stop when ace appears)
Use GEOMETRIC distribution (not binomial)

2. Complex Binomial Applications

Example 2: Quality Control Problem

Scenario: A factory produces computer chips. Quality control tests show 5% are defective. A batch of 50 chips is randomly selected for inspection.

Questions to answer:

What's the probability of finding exactly 3 defective chips?
What's the probability of finding at most 2 defective chips?
How many defects do we expect in a batch of 50?

Setup: n = 50, p = 0.05 (5% defect rate)

Answer 1: Exactly 3 defects

P(X = 3) = C(50,3) × (0.05)³ × (0.95)⁴⁷

= 19,600 × 0.000125 × 0.0934

≈ 0.228 or 22.8%

Answer 2: At most 2 defects

P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2)

This requires multiple calculations. Using technology is recommended!

P(X ≤ 2) ≈ 0.542 or 54.2%

Answer 3: Expected defects

μ = np = 50 × 0.05 = 2.5 defects

σ = √(np(1-p)) = √(50 × 0.05 × 0.95) = √2.375 ≈ 1.54 defects

Example 3: Medical Treatment

Scenario: A new treatment is effective for 75% of patients. A clinic treats 20 patients.

Questions:

What's the probability at least 18 patients improve?
What's the probability fewer than 12 improve?

Setup: n = 20, p = 0.75

Answer 1: At least 18 improve

P(X ≥ 18) = P(X=18) + P(X=19) + P(X=20)

Using binomial formula for each:

P(X=18) = C(20,18) × (0.75)¹⁸ × (0.25)² ≈ 0.0669

P(X=19) = C(20,19) × (0.75)¹⁹ × (0.25)¹ ≈ 0.0211

P(X=20) = C(20,20) × (0.75)²⁰ × (0.25)⁰ ≈ 0.0032

P(X ≥ 18) ≈ 0.0669 + 0.0211 + 0.0032 = 0.091 or 9.1%

Answer 2: Fewer than 12 improve

P(X < 12) = P(X ≤ 11) - calculated using technology ≈ 0.073 or 7.3%

3. Using Technology for Binomial Calculations

For large n or complex probability questions (like P(X ≥ k)), manual calculation becomes tedious. Technology makes these calculations much faster and more accurate.

Calculator and Software Functions

Tool	Function	Example
TI-84 Calculator	binompdf(n, p, k) for P(X=k) binomcdf(n, p, k) for P(X≤k)	binompdf(50, 0.05, 3) = 0.228
Excel	=BINOM.DIST(k, n, p, FALSE) for P(X=k) =BINOM.DIST(k, n, p, TRUE) for P(X≤k)	=BINOM.DIST(3, 50, 0.05, FALSE)
R	dbinom(k, n, p) for P(X=k) pbinom(k, n, p) for P(X≤k)	dbinom(3, 50, 0.05)
Python	scipy.stats.binom.pmf(k, n, p) scipy.stats.binom.cdf(k, n, p)	binom.pmf(3, 50, 0.05)

Tips for Using Technology:

Always verify inputs: check n, p, and k are correct
For P(X ≥ k), use: P(X ≥ k) = 1 - P(X ≤ k-1)
For P(a < X < b), use: P(X ≤ b-1) - P(X ≤ a)
Check answers make sense (probabilities between 0 and 1)

4. Normal Approximation to Binomial

When n is large and p is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution. This makes calculations even easier!

Conditions for Normal Approximation

The binomial distribution can be approximated by normal if:

np ≥ 10 AND n(1-p) ≥ 10

If conditions are met, use:
X ~ N(μ = np, σ = √(np(1-p)))

Example 4: Normal Approximation

Question: Flip a fair coin 100 times. Approximate P(X ≥ 60 heads).

Check conditions:

np = 100(0.5) = 50 ≥ 10

n(1-p) = 100(0.5) = 50 ≥ 10

Normal approximation is appropriate!

Parameters:

μ = np = 50

σ = √(np(1-p)) = √25 = 5

Approximation: X ~ N(50, 5)

P(X ≥ 60) ≈ P(Z ≥ (60-50)/5) = P(Z ≥ 2.0) ≈ 0.023

Interpretation: Only about 2.3% chance of getting 60+ heads in 100 flips.

When NOT to use normal approximation:

If np < 10 or n(1-p) < 10, the approximation is poor. Use the exact binomial formula or technology instead. The binomial distribution is too skewed when p is extreme.

5. Preview: Other Discrete Distributions

Poisson Distribution

Used for counting the number of rare events occurring in a fixed interval (time, space, volume). Examples: Number of emails per hour, accidents per month, defects per square meter.

Parameter: λ (lambda) = average number of events

Formula: P(X = k) = (e^(-λ) × λ^k) / k!

Geometric Distribution

Used when counting the number of trials needed to get the first success. Examples: Rolls until first 6, flips until first heads, calls until first sale.

Parameter: p = probability of success

Formula: P(X = k) = (1-p)^(k-1) × p

These distributions will be explored in more advanced statistics courses. For now, focus on mastering the binomial distribution!

Check Your Understanding

Question 1: Why is sampling without replacement NOT binomial?

Answer: When sampling without replacement, the probability of success changes after each draw. This violates the "constant probability" condition. For example, drawing aces from a deck: P(first ace) = 4/52, but P(second ace | first was ace) = 3/51.

Question 2: Can you use normal approximation for n=20, p=0.05? Why or why not?

Answer: No. Check: np = 20(0.05) = 1 < 10. The np ≥ 10 condition is violated. The distribution is too skewed when p is this small. Use exact binomial instead.

Question 3: Using a calculator, how would you find P(X ≥ 7) for a binomial with n=20, p=0.3?

Answer: Use the complement rule:

P(X ≥ 7) = 1 - P(X ≤ 6) = 1 - binomcdf(20, 0.3, 6)

On TI-84: 1 - binomcdf(20, 0.3, 6) ≈ 0.392

Question 4: What type of distribution models "number of phone calls received per hour"?

Answer: Poisson distribution. We're counting rare events (phone calls) in a fixed time interval (per hour), with no fixed number of "trials." This doesn't fit binomial conditions.

Summary

Key Takeaways:

Choose binomial when all four FTIC conditions are met
Use technology (calculators, Excel, R, Python) for complex probability calculations
Normal approximation works when np ≥ 10 and n(1-p) ≥ 10
Other discrete distributions (Poisson, Geometric) apply when binomial conditions are violated
Always verify conditions before applying any distribution

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