Question

Probability played a role in the rigging of the April \(24,1980,\) Pennsylvania state lottery. To determine each digit of the three-digit winning number, each of the numbers \(0,1,2, \ldots, 9\) is written on a Ping-Pong ball, the 10 balls are blown into a compartment, and the number selected for the digit is the one on the ball that floats to the top of the machine. To alter the odds, the conspirators injected a liquid into all balls used in the game except those numbered 4 and \(6,\) making it almost certain that the lighter balls would be selected and determine the digits in the winning number. They then proceeded to buy lottery tickets bearing the potential winning numbers. How many potential winning numbers were there (666 was the eventual winner)?

Short answer

Answer: There are 8 potential winning numbers for a three-digit combination containing only numbers 4 and 6.

Step-by-step solution

Find possible combinations for each digit

Since the winning number is a three-digit number and we know that only balls 4 and 6 are likely to be selected as the digits, we have 2 possibilities for each digit, either a 4 or a 6.

Apply the counting principle

To find the total number of possible winning numbers, we can use the counting principle. The counting principle states that if an operation can be performed in m ways and another operation can be performed in n ways, then the number of ways both operations can be performed is m*n. Here, all three digits have 2 possibilities, so the total number of potential winning numbers will be: 2 (1st digit) * 2 (2nd digit) * 2 (3rd digit) = 8 potential winning numbers. Note: The actual winning number was 666.

Key concepts

Counting Principle

The counting principle is a fundamental concept in probability and combinatorics.
It helps us determine the number of possible outcomes when there are multiple stages in a process.
If there are several actions that can be performed, this principle allows us to calculate the total number of ways these actions can take place.

• When one action can be performed in \(m\) ways.
• And another action in \(n\) ways.
• The total number of ways to perform both actions together is \(m \times n\).

In the context of our lottery example, each digit of the winning number could be either a 4 or a 6.
Therefore, there are 2 possibilities for each digit.
So, according to the counting principle, with three digits, the total number of possible winning numbers would be \(2 \times 2 \times 2 = 8\).
Understanding the counting principle is crucial in many applications, not just in gambling scenarios but also in fields like operations research and computer science.

Combinatorics

Combinatorics is the field of mathematics that deals with counting, arranging, and analyzing sets of elements.
It involves the combination and selection of objects.
Combinatorics allows us to determine how certain selections can be made from various groups of items.

Why is Combinatorics Important?

• Helps solve problems related to counting and probability.
• Essential in understanding complex systems and algorithms that depend on combinations.
• Used in many real-world applications including cryptography and network theory.

For example, in our lottery scenario, combinatorics helps us understand how many combinations of lottery numbers are possible.
This ensures a clear analysis of how the arrangement of numbers can result from the selection process.
By applying these principles, we found there are 8 possible outcomes when restricted choices for each digit were provided.

Lottery Rigging

Lottery rigging involves tampering with the process of selecting winning numbers to favor specific outcomes.
In the given scenario, the conspiring individuals injected liquid into unwanted balls to make the balls with desired numbers lighter.

How Lottery Rigging Works

• Altering physical characteristics of the objects used in drawing, like changing their weight.
• Ensuring specific numbers have a higher probability of being selected.
• Manipulating the random process to guarantee predictable results.

In this case, they left only balls numbered 4 and 6 light enough to float, skewing the selection process.
This unethical act highlights the importance of strict oversight and transparency in lottery games.
Rigging not only damages the integrity of the game but also has financial and legal implications.

Random Selection

Random selection is a process ensuring each option has an equal opportunity to be chosen.
It is a fundamental principle within probability theory and is crucial for fairness.

What Ensures True Randomness?

• No manipulation of the selection medium, such as weighted balls in a lottery.
• Consistent and transparent procedures observed in every instance of selection.
• Employing devices or algorithms designed to produce random outcomes.

In a fair lottery, each number from 0 to 9 should have an equal chance of being selected.
However, in the example provided, manipulation of the Ping-Pong balls compromised the randomness.
This emphasizes the need for well-regulated systems to maintain the integrity of games of chance and trust in their outcomes.