Question

In a binomial distribution consisting of five independent trails, the probability of 1 and 2 success are 0.4096 and 0.2048 respectively. Then, the parameter 'p' of distribution is

• A. \( \frac{1}{9} \)
• B. \( \frac{1}{7} \)
• C. \( \frac{1}{5} \)
• D. \( \frac{1}{2} \)

Hint:

For binomial (and Poisson) distribution problems where two probabilities \(P(X=k)\) and \(P(X=k+1)\) are given, taking their ratio is almost always the fastest way to solve for the parameter. This method cancels out most of the terms, leaving a simple linear equation.

Answer 1

The Correct Option is C

Step 1: Concept Explanation:
Given two probabilities from a binomial distribution, the goal is to determine the success probability, \(p\). This involves creating two equations based on the binomial probability formula and solving for \(p\).

Step 2: Core Formula:
The binomial probability mass function is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where \(n\) represents the number of trials, \(k\) represents the number of successes, and \(p\) represents the probability of success. A helpful strategy is to divide the two given probabilities to simplify the equations.

Step 3: Step-by-Step Solution:
We have:- Number of trials, \( n = 5 \).- \( P(X = 1) = 0.4096 \).- \( P(X = 2) = 0.2048 \).Formulate the equations:1. \( P(X=1) = \binom{5}{1} p^1 (1-p)^{5-1} = 5p(1-p)^4 = 0.4096 \)2. \( P(X=2) = \binom{5}{2} p^2 (1-p)^{5-2} = 10p^2(1-p)^3 = 0.2048 \)Divide equation (2) by equation (1):\[ \frac{P(X=2)}{P(X=1)} = \frac{10p^2(1-p)^3}{5p(1-p)^4} \]Substitute the given probability values:\[ \frac{0.2048}{0.4096} = \frac{1}{2} \]Simplify the expression:\[ \frac{10p^2(1-p)^3}{5p(1-p)^4} = \frac{2p}{1-p} \]Equate the two results:\[ \frac{2p}{1-p} = \frac{1}{2} \]Solve for \(p\):\[ 2 \times (2p) = 1 \times (1-p) \]\[ 4p = 1 - p \]\[ 5p = 1 \]\[ p = \frac{1}{5} \]
Step 4: Final Result:
The value of 'p' for this binomial distribution is \( \frac{1}{5} \).