Compute Binomial Probabilities in C

Introduction

In this lab, we will learn how to compute binomial probabilities in C programming language. The lab covers the following steps:

1. Reading the input values for the binomial probability calculation, including the number of trials (n), the probability of success (p), and the number of successes (k).
2. Implementing functions to calculate the factorial and combinations (n choose k), which are used in the binomial probability formula.
3. Calculating the binomial probability using the formula P(X=k) = C(n,k) _ p^k _ (1-p)^(n-k), where C(n,k) represents the combinations.
4. Printing the calculated probability.

By the end of this lab, you will have a solid understanding of how to work with binomial probabilities in C and be able to apply these concepts to various real-world problems.

Skills Graph

Read n, p, k

In this step, we will learn how to read input values for binomial probability calculation: n (number of trials), p (probability of success), and k (number of successes).

First, let's create a C program to read these input values:

#include <stdio.h>

int main() {
    int n, k;
    double p;

    printf("Enter number of trials (n): ");
    scanf("%d", &n);

    printf("Enter probability of success (p): ");
    scanf("%lf", &p);

    printf("Enter number of successes (k): ");
    scanf("%d", &k);

    printf("Input values:\n");
    printf("n = %d\n", n);
    printf("p = %.2f\n", p);
    printf("k = %d\n", k);

    return 0;
}

Let's save this file and compile it:

nano ~/project/binomial_prob.c
gcc ~/project/binomial_prob.c -o ~/project/binomial_prob

Example output when running the program:

Enter number of trials (n): 10
Enter probability of success (p): 0.5
Enter number of successes (k): 6
Input values:
n = 10
p = 0.50
k = 6

This program demonstrates how to:

1. Declare variables for n, p, and k
2. Use scanf() to read integer and floating-point inputs
3. Print the input values for verification

The input values represent:

• n: Total number of independent trials
• p: Probability of success in each trial
• k: Number of successful trials we want to calculate probability for

Compute P(X=k)=C(n,k)p^k(1-p)^(n-k)

In this step, we'll extend our previous program to compute the binomial probability using the formula P(X=k) = C(n,k) _ p^k _ (1-p)^(n-k).

We'll add functions to calculate combinations and binomial probability:

#include <stdio.h>
#include <math.h>

// Function to calculate factorial
unsigned long long factorial(int n) {
    if (n <= 1) return 1;
    return n * factorial(n - 1);
}

// Function to calculate combinations (n choose k)
unsigned long long combinations(int n, int k) {
    return factorial(n) / (factorial(k) * factorial(n - k));
}

// Function to calculate binomial probability
double binomial_probability(int n, int k, double p) {
    unsigned long long combinations_value = combinations(n, k);
    double probability = combinations_value *
                         pow(p, k) *
                         pow(1 - p, n - k);
    return probability;
}

int main() {
    int n, k;
    double p;

    printf("Enter number of trials (n): ");
    scanf("%d", &n);

    printf("Enter probability of success (p): ");
    scanf("%lf", &p);

    printf("Enter number of successes (k): ");
    scanf("%d", &k);

    double prob = binomial_probability(n, k, p);

    printf("Binomial Probability P(X=%d) = %f\n", k, prob);

    return 0;
}

Compile and run the program:

gcc ~/project/binomial_prob.c -o ~/project/binomial_prob -lm
~/project/binomial_prob

Example output:

Enter number of trials (n): 10
Enter probability of success (p): 0.5
Enter number of successes (k): 6
Binomial Probability P(X=6) = 0.205078

Key components of the binomial probability calculation:

1. factorial(): Calculates n!
2. combinations(): Calculates C(n,k) or number of ways to choose k items from n
3. binomial_probability(): Computes P(X=k) using the full formula
4. pow() function from math.h used for exponentiation

Print the Probability

In this step, we'll enhance our binomial probability program to provide more detailed and formatted output of the probability calculation.

We'll modify the previous program to include additional probability representations:

#include <stdio.h>
#include <math.h>

// Previous functions (factorial, combinations, binomial_probability) remain the same

int main() {
    int n, k;
    double p;

    printf("Enter number of trials (n): ");
    scanf("%d", &n);

    printf("Enter probability of success (p): ");
    scanf("%lf", &p);

    printf("Enter number of successes (k): ");
    scanf("%d", &k);

    double prob = binomial_probability(n, k, p);

    // Detailed probability output
    printf("\nProbability Calculation Results:\n");
    printf("-----------------------------\n");
    printf("Number of Trials (n):       %d\n", n);
    printf("Probability of Success (p): %.4f\n", p);
    printf("Number of Successes (k):    %d\n", k);

    // Different probability representations
    printf("\nProbability Representations:\n");
    printf("Decimal:     %f\n", prob);
    printf("Percentage:  %.2f%%\n", prob * 100);
    printf("Fraction:    1 in %.0f\n", 1.0 / prob);

    return 0;
}

Compile and run the program:

gcc ~/project/binomial_prob.c -o ~/project/binomial_prob -lm
~/project/binomial_prob

Example output:

Enter number of trials (n): 10
Enter probability of success (p): 0.5
Enter number of successes (k): 6

Probability Calculation Results:
-----------------------------
Number of Trials (n):       10
Probability of Success (p): 0.5000
Number of Successes (k):    6

Probability Representations:
Decimal:     0.205078
Percentage:  20.51%
Fraction:    1 in 5

Key improvements in this step:

1. Added detailed result formatting
2. Displayed multiple probability representations
3. Improved readability of output

Summary

In this lab, we learned how to read input values for binomial probability calculation, including the number of trials (n), probability of success (p), and number of successes (k). We also implemented functions to calculate factorial, combinations, and the binomial probability formula P(X=k) = C(n,k) _ p^k _ (1-p)^(n-k). This allows us to compute the probability of observing k successes in n independent trials, given the probability of success in each trial.

The key steps covered in this lab include reading input values, computing the binomial probability using the formula, and printing the resulting probability. This knowledge can be applied to various real-world scenarios where binomial probability is relevant, such as in statistics, decision-making, and risk analysis.