Question

In a fierce battle, not less than \(70 \%\) of the soldiers lost one eye, not less than \(75 \%\) lost one ear, not less than \(80 \%\) lost one hand, and not less than \(85 \%\) lost one leg. What is the smallest percentage who could have lost simultaneously one car, one eye, one hand, and one leg? This problem comes from Tangled Tales by Lewis Carroll, the author of Alice in Wonderland.

Short answer

At least 10% of soldiers lost an eye, an ear, a hand, and a leg.

Step-by-step solution

Define Events and Data

Let's define the events: Let \( E_1 \) be the event that a soldier lost one eye, \( E_2 \) the event that a soldier lost one ear, \( E_3 \) the event that a soldier lost one hand, and \( E_4 \) the event that a soldier lost one leg. We have: \( P(E_1) \geq 0.7 \), \( P(E_2) \geq 0.75 \), \( P(E_3) \geq 0.8 \), \( P(E_4) \geq 0.85 \).

Apply Inclusion-Exclusion Principle

To find the smallest percentage of soldiers who could have lost all four, we use the Inclusion-Exclusion Principle for 4 events: \[ P(E_1 \cap E_2 \cap E_3 \cap E_4) = 1 - (P(E_1') + P(E_2') + P(E_3') + P(E_4')). \] Where \( P(E_1') \leq 0.3 \), \( P(E_2') \leq 0.25 \), \( P(E_3') \leq 0.2 \), and \( P(E_4') \leq 0.15 \).

Calculate Complementary Probabilities

Calculate the sum of complementary probabilities: \[ P(E_1') + P(E_2') + P(E_3') + P(E_4') \leq 0.3 + 0.25 + 0.2 + 0.15 = 0.9. \]

Calculate the Intersection Probability

Substitute the result from Step 3 into the Inclusion-Exclusion Principle formula: \[ P(E_1 \cap E_2 \cap E_3 \cap E_4) = 1 - 0.9 = 0.1. \] This represents the minimum percentage of soldiers who lost an eye, an ear, a hand, and a leg.

Key concepts

Probability Theory

Probability theory is the mathematical framework for quantifying uncertainty and likelihood of events. It relies on the fundamental concepts of probability measures and random variables. When dealing with events, such as in our exercise concerning soldiers losing limbs and senses, we often focus on the probability of a single event occurring or unions and intersections of multiple events.
The probability of an event, say a soldier losing an eye (\( E_1 \), is denoted by \( P(E_1) \). This value ranges between 0 and 1, where 0 means the event is impossible and 1 means it is certain. In our example, we know \( P(E_1) \geq 0.7 \), meaning at least 70% of soldiers lost an eye.
Probability further extends to combining probabilities. We compute the likelihood of simultaneous events using principles like Inclusion-Exclusion, which allow us to adjust for over-counting occurrences that belong to multiple events.

Set Operations

Set operations are fundamental in understanding and solving problems that involve groups of items, particularly in probability and combinatorics. Sets are collections of distinct objects, which in probability, can represent events or outcomes.
In our battle scenario, each limb or sensory loss can be seen as separate events, represented as sets. For example, \( E_1 \) for losing an eye and \ \( E_2 \) for losing an ear. Set operations, such as union, intersection, and complement, help us solve problems involving these events.

• Union (\( E_1 \cup E_2 \)): Represents soldiers who lost an eye, an ear, or both. The union of two sets includes all elements from both sets.
• Intersection (\( E_1 \cap E_2 \)): Represents soldiers who lost both an eye and an ear. It's the overlapping portion of the sets.
• Complement (\( E_1' \)): The soldiers who did not lose an eye. It's calculated as 1 minus the probability of the event occurring.

These operations are the cornerstone of applying the Inclusion-Exclusion Principle, which is crucial for accurate probability calculations in the presence of overlapping sets.

Combinatorial Mathematics

Combinatorial mathematics focuses on counting, arranging, and optimizing elements within sets, which is essential for probability and statistical analysis. In our exercise, it helps us understand the combinations of possible outcomes in events.
Using combinatorics, we apply the Inclusion-Exclusion Principle to calculate how many soldiers simultaneously suffered various injuries. This principle considers overlapping group characteristics and provides an efficient way to determine the sum's true count.
The Inclusion-Exclusion Principle is expressed as:\[ P(E_1 \cap E_2 \cap E_3 \cap E_4) = 1 - (P(E_1') + P(E_2') + P(E_3') + P(E_4')) \]This formula effectively counts each individual's unique contribution by subtracting the probabilities of the complementary events.
In summary, combinatorial mathematics provides the logical structure and tools needed to efficiently manage multiple simultaneous conditions, which is essential for tackling the exercise at hand.