Question

A die is thrown 100 times. Getting an even number is considered a success. The variance of the number of successes is:

• A. 20
• B. 25
• C. 30
• D. 50

Hint:

For a binomial distribution where the probability of success is \(p = 0.5\), the variance simplifies to:
\[ \text{Variance} = \frac{n}{4} \]
With \(n = 100\), we calculate \(\frac{100}{4} = 25\) instantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem represents a series of independent, repeated Bernoulli trials, which is modeled using a binomial distribution.

Step 2: Key Formula or Approach:
For a binomial distribution with parameters \(n\) (number of trials) and \(p\) (probability of success):
\[ \text{Variance} = n p q \]
where \(q = 1 - p\) is the probability of failure.

Step 3: Detailed Explanation:
We are given:
- Number of times the die is thrown, \(n = 100\)
- A success is defined as rolling an even number.
A standard, fair six-sided die has the following outcomes:
\[ S = \{1, 2, 3, 4, 5, 6\} \]
The even numbers are:
\[ E = \{2, 4, 6\} \]
There are 3 successful outcomes out of 6 possible outcomes.
Therefore, the probability of success in a single trial, \(p\), is:
\[ p = \frac{3}{6} = \frac{1}{2} = 0.5 \]
The probability of failure, \(q\), is:
\[ q = 1 - p = 1 - 0.5 = 0.5 \]
Now, calculate the variance of the number of successes using our binomial formula:
\[ \text{Variance} = n p q \]
\[ \text{Variance} = 100 \times 0.5 \times 0.5 \]
\[ \text{Variance} = 100 \times 0.25 = 25 \]
The variance of the number of successes is 25.
This matches the second option.

Step 4: Final Answer:
Therefore, the correct option is (B).