Probability of Success in Binomial Distribution

In a binomial distribution, the probability of success (denoted as p) represents the likelihood of a specific event occurring in each trial. The binomial distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).

Given Information

In this case, we are provided with the following information:
- Number of independent trials (n) = 5
- Probability of 2 successes (P(X = 2)) = 0.4362
- Probability of 3 successes (P(X = 3)) = 0.2181

Calculating the Probability of Success (p)

To find the value of the parameter p, we can use the formula for the binomial probability mass function:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

where (n choose k) represents the binomial coefficient, which is calculated as:

(n choose k) = n! / (k! * (n - k)!)

Calculating the Probability of 2 Successes

Using the given information, we have:
P(X = 2) = 0.4362
n = 5

Substituting these values into the binomial probability formula, we get:

0.4362 = (5 choose 2) * p^2 * (1 - p)^(5 - 2)

Simplifying further:

0.4362 = 10 * p^2 * (1 - p)^3

Calculating the Probability of 3 Successes

Similarly, using the given information, we have:
P(X = 3) = 0.2181
n = 5

Substituting these values into the binomial probability formula, we get:

0.2181 = (5 choose 3) * p^3 * (1 - p)^(5 - 3)

Simplifying further:

0.2181 = 10 * p^3 * (1 - p)^2

Solving the Equations

We now have two equations with the unknown parameter p:

Equation 1: 0.4362 = 10 * p^2 * (1 - p)^3
Equation 2: 0.2181 = 10 * p^3 * (1 - p)^2

To solve these equations, we can use numerical methods or calculators. By solving the equations, we find that the value of p is approximately 2/3 or 0.6667.

Therefore, the correct answer is C) 2/3.