Sampling With vs. Without Replacement: The Key to Choosing the Right Denominator

[Rajeev Bagra]

🧩 Q1. Why was the denominator 4^7 in the birthday/season problem?

Question (from a student):
In the birthday-type problem, you used a denominator of 4^7 when calculating the probability that 7 people have birthdays in different seasons. But in the class-selection problem, you did not use 6^7. Why is the treatment different?

Answer:
Great question! The difference comes from the type of selection happening in each scenario.

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✔ Birthday/Season Problem

When 7 people each “choose a season” for their birthday, the choices behave like independent rolls of a 4-sided die:

• Each person chooses 1 of 4 seasons
• Selections can repeat
• Order matters
• Choices are independent

This is sampling with replacement.

So the number of total possible outcomes is:

4^7

That is why the denominator in that probability calculation is 4^7.

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🧩 Q2. Why is the class-selection problem different? Why can’t we use 6^7?

Question (from the same student):
In the class problem, Alice picks 7 classes from 30 (6 per weekday). Why is the denominator C(30,7) instead of 6^7? Isn’t this similar to the birthday case?

Answer:
It may look similar on the surface, but the underlying experiment is completely different.

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✔ Class-Selection Problem

Alice is registering for 7 distinct classes out of 30. This means:

• She cannot register for the same class twice
• Order does not matter (the set of 7 classes is what matters)
• These are not 7 independent choices
• This is sampling without replacement, and order does not matter

Therefore, the correct total number of possible outcomes is:

C(30,7)
(read as “30 choose 7”)

Using 6^7 would incorrectly assume:

• she can pick the same class more than once
• order matters
• choices are independent

These assumptions do not match the real situation.

So the denominator must be:

C(30,7)

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🧠 Summary: Why the denominators differ

Scenario	Replacement Allowed?	Order Matters?	Correct Denominator
Birthday/Seasons	Yes	Yes	4^7 (or 365^n for actual birthdays)
Class Selection	No	No	C(30,7)

Even though both problems involve “categories” like days or seasons, the probability model is completely different — and therefore the denominators must be different.

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