Choosing the right model

A probabilistic model is never "true": it is suited to the situation or it is not. Knowing how to choose it, and spotting when it does not fit, matters as much as knowing how to compute.

The three questions to ask

1. What are the OUTCOMES? (the sample space)
2. Are they EQUALLY LIKELY? (nothing forces this)
3. Is the experiment repeated WITH or WITHOUT replacement?

The third question changes the computations completely.

With or without replacement

WITH replacement                     WITHOUT replacement
---------------------------------    ---------------------------------
the drawn object is put back         the drawn object is kept
draws are INDEPENDENT                draws influence each other
the composition does not change      it changes at each draw

Example: 3 red balls, 2 black ones

P(2 reds) = (3/5) × (3/5) = 9/25     P(2 reds) = (3/5) × (2/4) = 3/10
                        = 0.36                              = 0.30

Without replacement, after drawing a red one only 2 reds remain out of 4 balls: the second probability changes. That is the difference between a lottery draw (without replacement) and a repeated die roll (with replacement).

Modelling a problem: a full example

An urn contains 4 tokens numbered 1, 2, 3, 4. Two are drawn simultaneously. What is the probability that their sum is even?

Step 1 — the sample space. A simultaneous draw is a draw without order and without replacement. The outcomes are the pairs:

{1,2} {1,3} {1,4} {2,3} {2,4} {3,4}        6 outcomes

Step 2 — equiprobability. The draw is "at random": the 6 pairs have the same probability, 1/6 each.

Step 3 — the event. The sum is even if both tokens have the same parity:

{1,3} -> 4   even    ✔
{2,4} -> 6   even    ✔
the other four give an odd sum

Step 4 — the computation.

P(even sum) = 2/6 = 1/3

The role of "at random"

The phrase "at random", in a problem, is what licenses equiprobability. It is never decorative: without it, the formula favourable / possible does not apply.

It can even be ambiguous. "Draw a chord at random in a circle" admits several legitimate models giving different answers: this is Bertrand's paradox, a reminder that probability depends on the protocol, not only on the figure.

What modelling does not guarantee

A probabilistic model describes an assumed mechanism. If the real mechanism differs — a slightly unbalanced die, a biased survey sample, an unwarranted independence assumption — the computations remain correct but the conclusions are false.

That is why, in statistics, a model's fit to the data is tested rather than taken for granted.

Summary

• Modelling means fixing the sample space, equiprobability and the protocol.
• With replacement: independent draws, unchanged composition.
• Without replacement: each draw modifies the following ones.
• A simultaneous draw is one without order or replacement: the outcomes are pairs.
• "At random" is what justifies equiprobability — never a decorative word.
• A badly chosen model gives exact computations and false conclusions.