Dice and dollars You roll a die, winning nothing if the...

Dice and dollars You roll a die, winning nothing if the number of spots is odd, $\$ 1$ for a 2 or a 4, and $\$ 10$ for a 6. a) Find the expected value and standard deviation of your prospective winnings. b) You play twice. Find the mean and standard deviation of your total winnings. c) You play 40 times. What’s the probability that you win at least $\$ 100$?

Dice and dollars You roll a die, winning nothing if the number of spots is odd, $\$ 1$ for a 2 or a 4, and $\$ 10$ for a 6.
a) Find the expected value and standard deviation of your prospective winnings.
b) You play twice. Find the mean and standard deviation of your total winnings.
c) You play 40 times. What’s the probability that you win at least $\$ 100$?

Key Concepts

Discrete Random Variables and Their Distributions

A discrete random variable is one that can take on a countable number of distinct outcomes, each with an associated probability. In problems of this type, a function assigns numerical values (such as winnings) to outcomes (such as the number on a die), and the probabilities for these outcomes form a probability distribution. Understanding this distribution is the first step in computing measures like the expected value and variance.

Expected Value

The expected value is a measure of the central tendency of a random variable, summarizing the average outcome if an experiment were repeated many times. It is computed by taking the sum of the products of each outcome and its respective probability. This concept is central to decision making under uncertainty and is used to predict long-term winnings or losses in games of chance.

Variance and Standard Deviation

Variance and standard deviation are measures of dispersion that describe the variability or spread of a random variable around its mean. Variance measures the average squared deviation from the expected value, while standard deviation is its square root, providing a scale similar to that of the original random variable. These concepts are essential for understanding the risk or uncertainty associated with the outcomes.

Linear Properties of Expectation for Independent Trials

When an experiment is repeated multiple times independently, the expected total outcome is simply the sum of the expected values of each individual trial due to the linearity of expectation. This property makes it straightforward to analyze aggregate results over several independent plays or trials.

Summing Variances for Independent Random Variables

For independent random variables, the variance of their sum is the sum of their individual variances. This property is crucial when computing the variability of total winnings over multiple independent plays, as it allows one to quantify the overall risk by simply adding up the variances of each play.

Central Limit Theorem and Normal Approximation

The Central Limit Theorem (CLT) states that the sum (or average) of a large number of independent, identically distributed random variables tends toward a normal distribution, regardless of the original distribution of the variables. This approximation enables the estimation of probabilities, such as the chance of reaching a certain total winnings, even when the original distribution is not normal.

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