A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ² de

Q: Wait, why is $X$ only 0 or 1? Can it be 2 or 3?

That is a fantastic question! Let us look at the pattern 'HT'. To have $X=2$, we would need the pattern 'HT' to appear twice, which would require at least four tosses (e.g., 'HTHT'). Since we only have three tosses, the maximum number of times 'HT' can appear is indeed 1. So, $X$ can only take the values 0 or 1.

Q: Why use the formula $\sigma^{2} = E[X^{2}] - (E[X])^{2}$ instead of $\sum (x_{i} - \mu)^{2} P(x_{i})$?

Both formulas are mathematically equivalent and will give you the same result! However, the formula $\sigma^{2} = E[X^{2}] - (E[X])^{2}$ is often faster and less prone to arithmetic errors because it avoids calculating the squared difference for each term individually. It is a powerful tool in your JEE toolkit.

Q: Is there a shortcut to finding the sample space for coin tosses?

Yes! For $n$ tosses, the number of outcomes is always $2^{n}$. For $n=3$, it is $2^{3}=8$. You can systematically list them by using a tree diagram or by writing them in binary (000, 001, 010, etc., where H=1 and T=0). This ensures you never miss an outcome.

Question

A coin is tossed three times. Let X denote the number of times a tail follows a head. If $\mu$ and $\sigma^{2}$ denote the mean and variance of X, then the value of $64(\mu+\sigma^{2})$ is:

Options:

(A) 51

(B) 64

(C) 32

(D) 48

Sigma Percentile · Accepted Answer

### Visualizing the Sample Space

When you toss a coin three times, each toss has two possibilities: Heads ($H$) or Tails ($T$). With three tosses, the total number of unique outcomes is $2^{3} = 8$.

The sample space $S$ is defined as:
$$S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$$

This represents the complete set of all possible outcomes for the experiment.

### The Hunt for the Pattern

We define the random variable $X$ as the number of times a tail follows a head. We are essentially hunting for the specific sequence '$HT$'.

Scanning our sample space:
*   $HHH$: $X=0$
*   $HHT$: $X=1$
*   $HTH$: $X=1$
*   $HTT$: $X=1$
*   $THH$: $X=0$
*   $THT$: $X=1$
*   $TTH$: $X=0$
*   $TTT$: $X=0$

The pattern '$HT$' appears in exactly four outcomes: $\{HHT, HTH, HTT, THT\}$. In these cases, $X=1$. In the other four outcomes, the pattern is absent, so $X=0$.

### The Probability Distribution

Since each of the 8 outcomes is equally likely, the probability of any single outcome is $\frac{1}{8}$.

The probability distribution is:
$$P(X=0) = \frac{4}{8} = \frac{1}{2}$$
$$P(X=1) = \frac{4}{8} = \frac{1}{2}$$

### Calculating Mean and Variance

We calculate the mean, $\mu = E[X]$, using the formula $\mu = \sum x_{i} P(X=x_{i})$:
$$\mu = 0 \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} = \frac{1}{2}$$

Next, we find the variance, $\sigma^{2}$, using the formula $\sigma^{2} = E[X^{2}] - (E[X])^{2}$. First, we calculate the expected value of the square:
$$E[X^{2}] = 0^{2} \cdot \frac{1}{2} + 1^{2} \cdot \frac{1}{2} = \frac{1}{2}$$

Now, we compute the variance:
$$\sigma^{2} = \frac{1}{2} - \left(\frac{1}{2}\right)^{2} = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$$

### The Final Evaluation

We are tasked with computing the value of $64(\mu + \sigma^{2})$. Substituting our calculated values:
$$64 \left( \frac{1}{2} + \frac{1}{4} \right)$$

Simplifying the expression inside the parentheses:
$$\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$$

Finally, we perform the multiplication:
$$64 \cdot \frac{3}{4} = 16 \cdot 3 = 48$$

The final result is **48**.

Animated Solution for Mathematics - Probability: A coin is tossed three times. Let X denote the number of times a tail follows a head. If $μ$ and $σ^{2}$ denote the mean and variance of X, then the value of $64 (μ + σ^{2})$ is:

Select Answer:

Visualized Solution

Defining the Sample Space

Defining Random Variable $X$

Mapping Outcomes to $X = 1$

Mapping Outcomes to $X = 0$

Probability Distribution of $X$

Formula for Mean $μ$

Calculating Mean $μ$

Formula for Variance $σ^{2}$

Calculating $E [X^{2}]$

Calculating Variance $σ^{2}$

Setting up $64 (μ + σ^{2})$

Final Evaluation

The Sigma Insight: Random Variables and Probability Distributions

Visualizing the Sample Space

The Hunt for the Pattern

The Probability Distribution

Calculating Mean and Variance

The Final Evaluation

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Select Answer:

Visualized Solution

Defining the Sample Space

Defining Random Variable .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X

Mapping Outcomes to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X=1

Mapping Outcomes to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X=0

Probability Distribution of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } X

Formula for Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } μ

Calculating Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } μ

Formula for Variance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σ2

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } E[X2]

Calculating Variance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σ2

Setting up .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 64(μ+σ2)