"The only way to do great work is to love what you do" - Steve Jobs
This is a series of posts that will focus on solving the daily problems posed by the user @ydavgonzalez step by step and we will also teach the programming approach.
Today, we're going to delve into the world of probability and explore a classic problem: rolling ten dice and determining the likelihood of their sum being 40. This seemingly simple problem leads us to a fascinating journey through the realms of mathematics and programming.
The Math Behind the Magic
Let's break down the core concepts that will guide us:
Sample Space: The sample space encompasses all possible outcomes of an event. In this case, each die has six possible outcomes (1 to 6), and we are rolling ten dice. So, the total number of possible outcomes is 6^10, which is a whopping 60,466,176 possibilities!
Favorable Outcomes: Our goal is to find the number of outcomes where the sum of the ten dice equals 40. This requires careful enumeration, and we can use techniques like combinations and permutations to streamline the process.
Probability: The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. In our case, it's the number of ways to get a sum of 40 divided by 60,466,176.
Coding the Solution
Now, let's write some code to tackle this problem using JavaScript:
function calculateProbability(numberOfDice, targetSum) {
let totalOutcomes = Math.pow(6, numberOfDice);
let favorableOutcomes = 0;
// Recursive function to explore all possible dice combinations
function countCombinations(currentDice, currentSum) {
if (currentDice === numberOfDice) {
if (currentSum === targetSum) {
favorableOutcomes++;
}
return;
}
for (let i = 1; i <= 6; i++) {
countCombinations(currentDice + 1, currentSum + i);
}
}
countCombinations(1, 0);
return favorableOutcomes / totalOutcomes;
}
let probability = calculateProbability(10, 40);
console.log(The probability of getting a sum of 40 with 10 dice is: ${probability});
Explanation:
- We define a function calculateProbability that takes the number of dice and the target sum as input.
- It calculates the total number of possible outcomes (totalOutcomes).
- A recursive function countCombinations iterates through all possible dice combinations.
- If the current sum matches the target sum, it increments the count of favorableOutcomes.
- Finally, the probability is calculated by dividing favorableOutcomes by totalOutcomes.
Challenge Time!
Now, you can modify the code to:
- Calculate the probability for different numbers of dice and target sums.
- Explore ways to optimize the code for efficiency.
- Implement alternative approaches using techniques like dynamic programming.
Share your solutions and insights in the comments below! Let's see what innovative ways you can tackle this problem.
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