Question

Suppose a coin is tossed three times. Let \(x\) be a random variable whose value is 1 if the number of heads is divisible by 3 , and 0 otherwise. Set up the sample space for \(x\) and the associated probabilities. Find \(\hat{x}\) and \(\sigma\).

Short answer

Sample Space: \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}. Probabilities: \(P(x = 1) = \frac{1}{4}\), \(P(x = 0) = \frac{3}{4}\). \(\hat{x} = \frac{1}{4}\), \(\sigma = \frac{\sqrt{3}}{4}\).

Step-by-step solution

Identify the Sample Space

When a coin is tossed three times, the sample space consists of all possible outcomes. Each outcome can be represented as a sequence of 'H' (heads) and 'T' (tails). Thus, the sample space is: \[ S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \] There are 8 possible outcomes.

Determine the Value of x for Each Outcome

For each outcome in the sample space, count the number of heads and check if it is divisible by 3. If it is, then \(x = 1\); otherwise, \(x = 0\). The evaluations are as follows: \[ \begin{align*} HHH & : 3 \text{ heads } (3 \text{ is divisible by 3}) \rightarrow x = 1 \ HHT & : 2 \text{ heads } \rightarrow x = 0 \ HTH & : 2 \text{ heads } \rightarrow x = 0 \ HTT & : 1 \text{ head } \rightarrow x = 0 \ THH & : 2 \text{ heads } \rightarrow x = 0 \ THT & : 1 \text{ head } \rightarrow x = 0 \ TTH & : 1 \text{ head } \rightarrow x = 0 \ TTT & : 0 \text{ heads } (0 \text{ is divisible by 3}) \rightarrow x = 1 \end{align*} \]

Calculate the Associated Probabilities

There are 2 outcomes where \(x = 1\) (HHH, TTT) and 6 outcomes where \(x = 0\) (HHT, HTH, HTT, THH, THT, TTH). Since each outcome in the sample space is equally likely, each has a probability of \( \frac{1}{8} \). Therefore, the probabilities are: \[ P(x = 1) = \frac{2}{8} = \frac{1}{4} \] \[ P(x = 0) = \frac{6}{8} = \frac{3}{4} \]

Find the Expected Value (Mean) \(\hat{x}\)

The expected value or mean of \(x\), denoted by \(\hat{x}\), is calculated as: \[ \hat{x} = E(x) = \sum (x_i \cdot P(x_i)) \] Substituting the values, we get: \[ \hat{x} = 1 \cdot \frac{1}{4} + 0 \cdot \frac{3}{4} = \frac{1}{4} \]

Find the Variance \(\sigma^2\) and Standard Deviation \(\sigma\)

First, compute the variance \(\sigma^2\) using the formula: \[ \sigma^2 = E(x^2) - (E(x))^2 \] Since \(x = 1\) or \(x = 0\), \(x^2 = x\). Thus, \[ E(x^2) = \hat{x} = \frac{1}{4} \] Therefore, \[ \sigma^2 = \frac{1}{4} - \left(\frac{1}{4}\right)^2 = \frac{1}{4} - \frac{1}{16} = \frac{3}{16} \] Now the standard deviation \(\sigma\) is: \[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{3}{16}} = \frac{\sqrt{3}}{4} \]

Key concepts

Sample Space

In probability theory, the sample space is the set of all possible outcomes of an experiment. For anyone new to this, it's essential to think of the sample space as a complete list of everything that can happen.

For example, when you toss a coin three times, you can get a series of heads (H) and tails (T). The complete set of these sequences is called the sample space. Here, the sample space is represented as:

\[ S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \]
Each sequence is equally likely, and there are eight possible outcomes in total. This simple listing helps us understand all the possible results that can occur from our experiment.

Random Variable

A random variable is a variable whose values depend on the outcomes of a random phenomenon. It's a way to map the results of the sample space to numerical values.

In our exercise, we define a random variable, \( x \), that takes the value 1 if the number of heads is divisible by 3, and 0 otherwise. This means we're translating the sequences of heads and tails into numbers we can work with.

For example, in the outcome sequence 'HHH' (which has 3 heads), since 3 is divisible by 3, we assign \( x = 1 \). For most other sequences where the count of heads isn't divisible by 3, we assign \( x = 0 \). This method helps us use numerical techniques to analyze the results.

Expected Value

The expected value, or mean, of a random variable gives a measure of the center of the distribution of the values that the variable can take.

It is calculated using the formula:
\[ E(x) = \sum (x_i \cdot P(x_i)) \]
Here, \( x_i \) represents the possible values of \( x \), and \( P(x_i) \) represents the probability of \( x_i \).
For our random variable \( x \), the possible values are 1 and 0 with probabilities \( \frac{1}{4} \) and \( \frac{3}{4} \), respectively. Thus, the expected value \( \hat{x} \) is:
\[ \hat{x} = 1 \cdot \frac{1}{4} + 0 \cdot \frac{3}{4} = \frac{1}{4} \]
This tells us that, on average, we can expect our random variable \( x \) to have a value of \( \frac{1}{4} \).

Variance

Variance measures how much the values of a random variable differ from the expected value. Essentially, it tells us how spread out the values are.
To calculate variance \(\sigma^2\), we use the formula:
\[ \sigma^2 = E(x^2) - (E(x))^2 \]
Since the square of our random variable \( x \) is itself (as \( x = 0 \) or \( x = 1 \)), the expected value of \( x^2 \) is the same as the expected value of \( x \):
\[ E(x^2) = \hat{x} = \frac{1}{4} \]
Therefore, the variance is calculated as:
\[ \sigma^2 = \frac{1}{4} - \left(\frac{1}{4}\right)^2 = \frac{1}{4} - \frac{1}{16} = \frac{3}{16} \]
This result shows us how much our random variable values can deviate from the mean.

Standard Deviation

The standard deviation is the square root of the variance. It provides a measure in the same units as the original data, making it easier to interpret.
We calculate standard deviation \( \sigma \) with:
\[ \sigma = \sqrt{\sigma^2} \]

In our case, the variance \( \sigma^2 \) is \( \frac{3}{16} \). Therefore, the standard deviation \( \sigma \) is:

\[ \sigma = \sqrt{\frac{3}{16}} = \frac{\sqrt{3}}{4} \]

The standard deviation helps us understand the typical deviation from the expected value, giving a clearer picture of the data's spread and how much individual outcomes differ from the mean.