Probability of Selecting Cards with 4 as the Largest Value: A Comprehensive Analysis

Introduction

This article explores the probability of selecting three cards from a box containing cards numbered 1 to 5, such that 4 is the largest value selected. We will break down the process step-by-step, using combinations and probability theory to derive a solution. This detailed analysis aims to provide a clear understanding of the underlying concepts.

Understanding the Problem

We have a box containing 5 cards, numbered 1 to 5. The task is to select three cards at random without replacement. The goal is to determine the probability that 4 is the largest value among the selected cards.

Identifying the Total Number of Ways to Select 3 Cards

The total number of ways to choose 3 cards from 5 can be calculated using the combination formula, which is given by:

[ binom{n}{k} frac{n!}{k!(n-k)!} ]

Here, n is the total number of items (5 cards), and k is the number of items to choose (3 cards).

Calculation Steps

[ binom{5}{3} frac{5!}{3!(5-3)!} frac{5 times 4}{2 times 1} 10 ]

The 10 combinations of selecting 3 cards from the 5 are:

{1, 2, 3} {1, 2, 4} {1, 2, 5} {1, 3, 4} {1, 3, 5} {1, 4, 5} {2, 3, 4} {2, 3, 5} {2, 4, 5} {3, 4, 5}

Identifying the Favorable Outcomes

For 4 to be the largest value selected, the other two cards must come from the set {1, 2, 3}. The number of ways to choose 2 cards from 3 is:

[ binom{3}{2} frac{3!}{2!(3-2)!} 3 ]

The combinations of selecting 2 cards from {1, 2, 3} are:

{1, 2} {1, 3} {2, 3}

Calculating the Probability

The probability that 4 is the largest card selected is the ratio of the number of favorable outcomes to the total outcomes:

[ P(4 text{ as largest}) frac{text{Favorable ways}}{text{Total ways}} frac{3}{10} 0.3 ]

Alternative Method: Permutations and Factorials

An alternative approach involves calculating permutations and then dividing by the factorial of the number of selected cards to account for the order not mattering. The total permutations of selecting 3 cards from 5 are:

[ frac{5!}{2!} 5 times 4 times 3 60 ]

Since order does not matter, we divide by the factorial of the number of selected cards (3!):

[ frac{60}{3!} frac{60}{6} 10 ]

The number of ways to select 2 cards from {1, 2, 3} and include the card 4 is 3, as calculated earlier. Therefore, the probability remains:

[ P(4 text{ as largest}) frac{3}{10} 0.3 ]

Conclusion

The probability that 4 is the largest value selected when drawing three cards from a box containing cards numbered 1 to 5 is 0.3 or 30%. This analysis demonstrates the use of combinations and permutations in solving probability problems and provides a clear, step-by-step solution.

Keywords

probability, combinations, card selection