Meaning of Binomial Distribution

Before we move ahead to solve the problem, let's first understand the concept of binomial distribution.

A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where the probability of success remains the same in all the trials. The distribution can be represented by two parameters - n and p, where n is the number of trials and p is the probability of success in each trial.

The mean of a binomial distribution is given by the formula:

μ = n × p

The standard deviation of a binomial distribution is given by the formula:

σ = √(n × p × (1 - p))

Solving the Problem

Now, let's solve the given problem step by step.

Given,

Mean (μ) = 3

Standard deviation (σ) = 1.5

We know that,

μ = n × p

σ = √(n × p × (1 - p))

Let's substitute the given values in these formulas and solve for n and p.

From the first equation, we get:

p = μ / n

Substituting the value of μ and p, we get:

3 = n × (μ / n)

3 = μ

So, n = 3 / μ = 3 / 3 = 1

Now, let's substitute the values of n and μ in the second equation:

σ = √(n × p × (1 - p))

1.5 = √(1 × p × (1 - p))

Squaring both sides, we get:

2.25 = p × (1 - p)

2.25 = p - p^2

Rearranging, we get:

p^2 - p + 2.25 = 0

Solving this quadratic equation, we get:

p = 1.5, 0.5

Since the probability of success cannot be greater than 1, we reject the value p = 1.5.

Therefore, p = 0.5

Now, we know that n = 1 and p = 0.5.

The number of trials in a binomial distribution is given by n. Therefore, the answer is:

No. of trials (n) = 3 / μ = 3 / 3 = 1 / p = 1 / 0.5 = 2

But, the question asks for the number of trials in a binomial distribution having mean and SD as 3 and 1.5 respectively. Since we have found the values of n and p, we can check if they satisfy the given conditions.

Mean (μ) = n × p = 1 × 0.5 = 0.5 ≠ 3

Standard deviation (σ) = √(n × p × (1 - p)) = √(1 × 0.5 × 0.5) = 0.5 ≠ 1.5

Therefore, the values of n and p that we have found do not satisfy the given conditions.

We need to go back and check our calculations.

From the equation p^2 - p + 2.25 = 0, we get two values of p:

p = 1.5, -0.5

We reject the negative value of p.

Therefore, p =