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: Understanding the Concept:
The problem provides two probabilities from a binomial distribution and asks to find the success probability parameter, \(p\). We can set up two equations using the binomial probability formula and solve for \(p\).

Step 2: Key Formula or Approach:
The binomial probability mass function is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where \(n\) is the number of trials, \(k\) is the number of successes, and \(p\) is the probability of success. A useful technique is to take the ratio of the two given probabilities to simplify the equations.

Step 3: Detailed Explanation:
We are given: - Number of trials, \( n = 5 \). - \( P(X = 1) = 0.4096 \). - \( P(X = 2) = 0.2048 \). Let's write out the equations: 1. \( P(X=1) = \binom{5}{1} p^1 (1-p)^{5-1} = 5p(1-p)^4 = 0.4096 \) 2. \( P(X=2) = \binom{5}{2} p^2 (1-p)^{5-2} = 10p^2(1-p)^3 = 0.2048 \) Now, let's take the ratio of equation (2) to equation (1): \[ \frac{P(X=2)}{P(X=1)} = \frac{10p^2(1-p)^3}{5p(1-p)^4} \] Substitute the given probability values: \[ \frac{0.2048}{0.4096} = \frac{1}{2} \] Now, simplify the expression with the parameters: \[ \frac{10p^2(1-p)^3}{5p(1-p)^4} = \frac{2p}{1-p} \] Equating the two results: \[ \frac{2p}{1-p} = \frac{1}{2} \] Now, solve for \(p\): \[ 2 \times (2p) = 1 \times (1-p) \] \[ 4p = 1 - p \] \[ 5p = 1 \] \[ p = \frac{1}{5} \]
Step 4: Final Answer:
The parameter 'p' of the distribution is \( \frac{1}{5} \).