Key Concepts and Formulas

1. Binomial Distribution: A discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant for each trial. A random variable $X$ following a binomial distribution is denoted as $X \sim B(n, p)$, where $n$ is the number of trials and $p$ is the probability of success in a single trial.
2. Probability of Failure ($q$): If $p$ is the probability of success, then the probability of failure $q$ is $q = 1 - p$.
3. Variance of a Binomial Distribution: For a random variable $X$ following a Binomial Distribution $B(n, p)$, the variance of the distribution of success is given by the formula: $\text{Var}(X) = npq$

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Step-by-Step Solution

Step 1: Confirming it's a Binomial Distribution and Identifying $n$

We first need to verify if the given problem scenario fits the conditions for a Binomial Distribution and then identify the number of trials ($n$).

• Fixed Number of Trials ($n$): The problem states, "A dice is tossed $5$ times." This indicates a fixed number of trials. Therefore, $n = 5$.
• Independent Trials: Each toss of a dice is an independent event. The outcome of one toss does not influence the outcome of any other toss.
• Two Possible Outcomes: For each toss, we are interested in whether we get an "odd number" (defined as a success) or not (defined as a failure, which means getting an even number).
• Constant Probability of Success: The probability of getting an odd number on a standard six-sided dice remains the same for every toss.

Since all four conditions are met, we can confidently use the Binomial Distribution framework to solve this problem.

Step 2: Determining the Probability of Success ($p$) and Failure ($q$)

Next, we calculate the probability of success ($p$) and the probability of failure ($q$) for a single trial.

• Probability of Success ($p$): A "success" is defined as "getting an odd number" when a dice is tossed. When a standard six-sided dice is tossed, the sample space (all possible outcomes) is $\{1, 2, 3, 4, 5, 6\}$. The total number of possible outcomes is $6$. The outcomes that are considered "success" (getting an odd number) are $\{1, 3, 5\}$. The number of favorable outcomes is $3$. The probability of success ($p$) in a single trial is calculated as: $p = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{3}{6} = \frac{1}{2}$ So, $p = \frac{1}{2}$.

• Probability of Failure ($q$): The probability of failure ($q$) is the complement of the probability of success, meaning $q = 1 - p$. This is because a trial must result in either success or failure. $q = 1 - p = 1 - \frac{1}{2} = \frac{1}{2}$ So, $q = \frac{1}{2}$.

Step 3: Calculating the Variance of the Distribution

Now that we have identified all the necessary parameters, $n$, $p$, and $q$, we can apply the formula for the variance of a Binomial Distribution.

We have:

• Number of trials, $n = 5$
• Probability of success, $p = \frac{1}{2}$
• Probability of failure, $q = \frac{1}{2}$

The formula for the variance of a Binomial Distribution is: $\text{Variance} (X) = npq$ Substitute the values into the formula: $\text{Variance} (X) = 5 \times \frac{1}{2} \times \frac{1}{2}$ $\text{Variance} (X) = 5 \times \frac{1}{4}$ $\text{Variance} (X) = \frac{5}{4}$

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Common Mistakes & Tips

1. Verify Distribution Type: Always ensure the problem fits the conditions of a Binomial Distribution (fixed independent trials, two outcomes, constant $p$) before applying its formulas.
2. Accurate $p$ Calculation: Carefully determine the probability of success ($p$) for a single trial. For standard dice problems, $p$ can be $1/6$, $1/2$, $1/3$, etc., depending on the definition of success.
3. Variance vs. Standard Deviation: Pay close attention to whether the question asks for variance ($npq$) or standard deviation ($\sqrt{npq}$).
4. Arithmetic Precision: Even simple calculations can lead to errors. Double-check your arithmetic, especially under exam conditions.

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Summary

The problem describes a classic scenario for a Binomial Distribution. We identified the number of trials $n=5$, and the probability of success $p=1/2$ (for getting an odd number on a die). Consequently, the probability of failure $q=1-p=1/2$. Using the formula for the variance of a Binomial Distribution, $\text{Var}(X) = npq$, we calculated the variance to be $5 \times \frac{1}{2} \times \frac{1}{2} = \frac{5}{4}$.

The final answer is $\boxed{\text{5/4}}$ which corresponds to option (D).