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 \(\bar{x}\) and \(\sigma\).

Short answer

The sample space for x is: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. \(\bar{x}\) = 1/8, \(\sigma = 0.331\).

Step-by-step solution

Determine the Sample Space

List all possible outcomes when a coin is tossed three times. These outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Define the Random Variable x

The random variable x takes the value 1 if the number of heads in any outcome is divisible by 3, otherwise it takes the value 0.

Identify Outcomes that Fit the Condition

Identify which outcomes result in a number of heads that is divisible by 3. There is only one such outcome: HHH, which has 3 heads.

Assign x Values to Each Outcome

Assign 1 to HHH since it has 3 heads (3 is divisible by 3). Assign 0 to all other outcomes (HHT, HTH, HTT, THH, THT, TTH, TTT) since their number of heads is not divisible by 3.

Construct the Probability Distribution

The probabilities for each outcome are uniform since each coin toss is fair. Each outcome has a probability of \(\frac{1}{8}\). Define the probability distribution for x: P(x=1) = 1/8, P(x=0) = 7/8.

Calculate Expected Value \(\bar{x}\)

The expected value \(\bar{x}\) can be calculated as \(\bar{x} = 1 \cdot \frac{1}{8} + 0 \cdot \frac{7}{8} = \frac{1}{8}\).

Calculate the Variance \( \sigma^2\)

To find the variance, first calculate E(x^2). Since x can only be 0 or 1, \(\bar{x^2}\) is the same as \(\bar{x}\). Then use the formula: \(\sigma^2 = E(x^2) - [E(x)]^2 = \frac{1}{8} - (\frac{1}{8})^2\ = 0.109375\).

Calculate the Standard Deviation \( \sigma\)

The standard deviation is the square root of the variance: \(\sigma = \sqrt{0.109375} \approx 0.331\).

Key concepts

random_variable

A random variable is a numerical outcome of a random process. It maps every outcome in the sample space to a numerical value. In our exercise, the random variable x takes the value 1 if the number of heads from tossing a coin three times is divisible by 3, and 0 otherwise. This way, each outcome of the coin tosses (like HHH, HHT, etc.) is assigned a specific value by the random variable x. For example, HHH results in x=1 because 3 (the number of heads) is divisible by 3, while HHT results in x=0 because 2 (the number of heads) is not divisible by 3.

expected_value

The expected value (E(x)), often written as \(\bar{x}\), is the average value of a random variable over many trials of the experiment. It represents the long-term average if the experiment is repeated many times. For our exercise, the expected value is calculated by weighting each value the random variable x can take (0 and 1) by its probability and summing the results. Specifically: \[\bar{x} = 1 \times \frac{1}{8} + 0 \times \frac{7}{8} = \frac{1}{8}\] This tells us that if we repeated the coin tossing many times, the average of the values taken by the random variable x would be 0.125.

variance

Variance (\(\sigma^2\)) measures how much the values of a random variable x differ from the expected value. It gives us an idea of the 'spread' or dispersion of x around \(\bar{x}\). In simple terms, it tells us if the values tend to be close to the expected value or far away from it. For our exercise, since x can only be 0 or 1, the calculations simplify a bit: first, we calculate E(x²), which in this case is the same as E(x) since x can only be 0 or 1. The formula for variance is then: \[\sigma^2 = E(x^2) - [E(x)]^2 = \frac{1}{8} - \[ \left( \frac{1}{8} \right \]^2 = 0.109375 \] Thus, the variance of x in our coin-toss experiment is 0.109375.

standard_deviation

Standard deviation (\(\sigma\)) is the square root of the variance and provides a measure of the average distance of each data point from the mean (expected value). It is a more intuitive measure of spread because it is in the same units as the random variable x. In our exercise, with a variance of 0.109375, the standard deviation is: \[\sigma = \sqrt{0.109375} \approx 0.331 \] This means that on average, the values of x (whether 0 or 1) deviate from the mean (\(\bar{x} \)) by about 0.331. It's a useful measure to understand the dispersion of x around its expected value.