The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respective

Q: Wait, why do we use the quadratic equation method instead of just substituting $v = 82.5 - m$ into the product equation?

You absolutely could! Substitution is a valid path. However, the quadratic method ($t^2 - (sum)t + product = 0$) is the 'JEE way'—it's faster, cleaner, and less prone to arithmetic errors. In high-stakes exams, elegance is a defense mechanism against mistakes.

Q: How do I know for sure that the mean is the larger value? Is it possible for variance to be larger?

Great question! In a binomial distribution, $v = npq$. Since $m = np$, we have $v = m \times q$. Because $q$ is a probability (the probability of failure), it must satisfy $0 \le q \le 1$. Therefore, $v$ must be less than or equal to $m$. If you ever calculate a variance larger than the mean, you know you've swapped your roots!

Q: What if I get a non-integer value for $n$? Does that mean I made a mistake?

Yes, absolutely. The number of trials $n$ represents a count of discrete events. It must be a positive integer. If your calculation yields a fraction, re-check your arithmetic or your assignment of the mean and variance.

Question

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. They the number of trials in the binomial distribution is:

Sigma Percentile · Accepted Answer

### Analyzing the Setup

In a binomial distribution, we deal with two fundamental parameters: the mean $m = np$ and the variance $v = npq$. These are the DNA of the distribution, where the mean represents the expected outcome and the variance measures the fluctuation.

We are given the following system of equations:
$$m + v = 82.5$$
$$mv = 1350$$

### The Algebraic Shortcut

While substitution is possible, we can utilize the elegance of Vieta's formulas. If $m$ and $v$ are the roots of a quadratic equation, that equation takes the form $t^2 - (m+v)t + mv = 0$.

Substituting our known values, we obtain:
$$t^2 - 82.5t + 1350 = 0$$

To simplify the arithmetic and clear the decimal, we multiply the entire equation by $2$:
$$2t^2 - 165t + 2700 = 0$$

### Solving the Quadratic

We now factorize the quadratic equation $2t^2 - 165t + 2700 = 0$. We seek two numbers that multiply to $5400$ and add to $-165$, which are $-120$ and $-45$.

Rewriting the equation:
$$2t^2 - 120t - 45t + 2700 = 0$$

Factoring by grouping:
$$2t(t - 60) - 45(t - 60) = 0$$
$$(2t - 45)(t - 60) = 0$$

This yields two potential roots: $t = 22.5$ and $t = 60$.

### The Critical Insight

We must determine which value corresponds to the mean and which to the variance. Recall that $v = mq$, where $q$ is the probability of failure.

Since $0 \le q \le 1$, it follows that $v \le m$. Therefore, we must assign the values as follows:
$$m = 60$$
$$v = 22.5$$

### Final Calculation

Now that we have identified $m$ and $v$, we calculate $q$:
$$q = \frac{v}{m} = \frac{22.5}{60} = \frac{3}{8}$$

Since $p + q = 1$, we find $p$:
$$p = 1 - \frac{3}{8} = \frac{5}{8}$$

Finally, we use the definition of the mean $m = np$ to solve for $n$:
$$60 = n 	imes \frac{5}{8}$$
$$n = \frac{60 	imes 8}{5} = 96$$

The number of trials is **$n = 96$**.

Animated Solution for Mathematics - Probability: The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. They the number of trials in the binomial distribution is:

Enter Numerical Value:

Visualized Solution

Define Mean and Variance

Translate the Problem into Equations

Form the Quadratic Equation

Simplify the Quadratic Equation

Factorize and Solve

Identify Mean and Variance

Calculate Probability of Failure $q$

Calculate Probability of Success $p$

Find the Number of Trials $n$

Conclusion and Key Takeaway

The Sigma Insight: Binomial Distribution

Analyzing the Setup

The Algebraic Shortcut

Solving the Quadratic

The Critical Insight

Final Calculation

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