The Probability of Picking a Penny from a Dollar Coin Stack

This simple question—if you randomly pick one of 50 coins that total 1 dollar, what is the probability that you pick a penny?—reveals the intricacies involved in calculating probabilities based on specific constraints. Let's delve into the detailed steps and analysis to solve this problem.

Understanding the Constraints

The given constraints are crucial in determining the possible combinations of coins. The total value of 1 dollar does not allow all the coins to be pennies, nickels, dimes, or quarters. The combinations must adhere to the following:

No more than one 50-cent piece No more than three 25-cent (quarter) pieces No more than eight 10-cent (dime) pieces No more than four 5-cent (nickel) pieces

Exploring Possible Combinations

Given the constraints, we explore possible combinations to meet the total value of 1 dollar with 50 coins:

1. One 50-cent Piece

Using one 50-cent piece, we need another 50 cents from 49 pennies. This is one valid combination:

45 pennies, 1 nickel, 2 dimes, and 1 quarter

2. No 50-cent Piece

Without any 50-cent pieces, we explore combinations using 25 cents, 10 cents, and 5 cents, ensuring the total is exactly 1 dollar and remains 50 coins:

40 pennies, 8 nickels, 2 dimes

These are the only two combinations that meet the criteria:

45 pennies, 1 nickel, 2 dimes, and 1 quarter 40 pennies, 8 nickels, 2 dimes

Calculating the Probabilities

For each combination, the probability of picking a penny is the number of pennies divided by the total number of coins (50).

First Combination

Number of pennies: 45 Total number of coins: 50 Probability: 45/50 9/10

Second Combination

Number of pennies: 40 Total number of coins: 50 Probability: 40/50 4/5

Determining the Final Probability

Without further information about the specific combination, we must assume that we are equally likely to pick either combination. Thus, the probability is the average of the two calculated probabilities:

Probability (4/5 9/10) / 2 8/10 9/10 / 2 17/20

Therefore, the final probability is 17/20, or 85%.

Conclusion

This analysis explicitly shows how to approach the problem systematically, ensuring that all constraints are met while calculating the probability accurately. Understanding these steps can help in tackling similar probability problems involving multiple variables and constraints.

Keywords: probability, coin stack, dollar value