Computation of Binomial Probabilities: A Step-by-Step Guide

Binomial probabilities are a core concept in statistics, widely used in various applications from quality control to genetics. This article provides a detailed guide on how to compute binomial probabilities directly from the formula, specifically focusing on the P(X leq; 5) calculation for n8 and p0.4.

Understanding Binomial Probability

The binomial probability formula is an essential tool in probability theory. It is used to calculate the probability of obtaining exactly x successes in n independent Bernoulli trials, where each trial has a success probability of p. The formula is:

P(Xx) C(n, x) * p^x * (1-p)^(n-x)

Here, C(n, x) or binom{n}{x} represents the binomial coefficient, which is the number of ways to choose x successes from n trials without regard to order.

Computation for Specific Values

In this section, we will calculate the binomial probabilities for n8 and p0.4, specifically for P(X leq; 5).

Step 1: Calculate P(X3)

First, we need to calculate the probability of exactly 3 successes:

Calculate the binomial coefficient:

C(8, 3) 8! / (3! * 5!) 56

Calculate p^x and (1-p)^(n-x):

0.4^3 0.064

0.6^5 0.07776

Combine these values:

P(X3) 56 * 0.064 * 0.07776 approx; 0.2234

Step 2: Calculate P(X2), P(X3), and P(X4)

Next, we calculate the probabilities for x2, x3, and x4:

P(X2):

C(8, 2) 8! / (2! * 6!) 28

0.4^2 * 0.6^6 approx; 0.2109

P(X3) (already calculated):

P(X3) approx; 0.2234

P(X4):

C(8, 4) 8! / (4! * 4!) 70

0.4^4 * 0.6^4 approx; 0.2335

Step 3: Combine the Probabilities

Finally, we combine the probabilities for x2, x3, and x4 to find P(X leq; 5):

P(X leq; 5) P(X2) P(X3) P(X4) approx; 0.2109 0.2234 0.2335 approx; 0.6678

Binomial Density Table

To further verify the calculations, we can refer to the binomial density table for n8 and p0.4 with x ranging from 0 to 8.

xPX x 00.01680 10.08958 20.20902 30.27869 40.23224 50.12386 60.04129 70.00786 80.00066

Conclusion

Through the detailed computations, we have successfully calculated the binomial probability P(X leq; 5) for n8 and p0.4. This guide not only provides a practical understanding of the binomial probability formula but also highlights the importance of accurate and systematic calculations in probability theory.