Birthday

This program calculates the “Birthday Problem” probabilities, determining how many people are needed in a group before there is at least a 50% chance that two or more share the same birthday. It iterates from 1 to 60 people, printing each group size alongside the running probability that at least two share a birthday. The probability complement is computed iteratively: Q accumulates the product of (365−N)/365 terms, representing the chance that all birthdays are distinct, and then (Q−1)×−100 converts this to a percentage chance of a collision. The SCROLL command is used to prevent the display from pausing, allowing all 60 results to stream continuously past.

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Program Analysis

Program Structure

The program is a straightforward iterative calculation across six lines. It initialises a probability accumulator, loops from 1 to 60, and prints results continuously using SCROLL to avoid the display filling up.

Line	Purpose
10	Prints a multi-line description of the problem
20	Initialises Q to 1 (the probability all birthdays are unique, starting condition)
30	Opens the main loop, N = group size from 1 to 60
40	SCROLL prevents the display from pausing between lines
50	Prints N and the collision probability as a percentage
60	Updates Q by multiplying in the next distinct-birthday factor
70	Closes the loop

Mathematical Approach

The Birthday Problem is solved using the complementary probability. Q holds the probability that all N people have distinct birthdays, computed as the running product:

Q = (365/365) × (364/365) × (363/365) × … × ((365−N+1)/365)

At each step, line 50 prints -(100*(Q-1)), which is equivalent to 100*(1-Q) — converting the complement probability into a percentage chance that at least two people share a birthday. The negation trick avoids the need for a separate subtraction from 1, since Q−1 is always negative or zero.

The classic result — that a group of 23 people suffices for a greater than 50% collision probability — will be clearly visible in the output around N=23.

Key BASIC Idioms

• SCROLL at line 40 suppresses the scroll prompt, allowing all 60 lines of output to flow without user intervention.
• The use of commas in the PRINT statement at line 10 advances to successive print zones, indenting the title text.
• The expression -(100*(Q-1)) is a compact idiom for 100*(1-Q), exploiting unary negation rather than rewriting the subtraction.
• Q is updated in place each iteration rather than recomputed from scratch, making this an efficient incremental calculation.

Notable Techniques

The iterative update of Q at line 60 — multiplying by (365-N)/365 — means the program never needs to compute a factorial or large product from scratch, keeping numerical operations within manageable floating-point range for all 60 iterations.

Bugs and Anomalies

The loop begins at N=1, for which the collision probability is 0% (one person cannot share a birthday with anyone else). This is mathematically correct but slightly redundant as a starting point. The loop runs to N=60, well past the point (~23) where probability exceeds 50%, which is useful for demonstrating how quickly the probability approaches certainty near N=60 (approximately 99.4%).

The title string on line 10 is not line-wrapped with explicit formatting — the text runs on as a single long string and relies on the display’s automatic wrapping, which may produce ragged output depending on screen width.