Seven thousand lottery tickets are sold for 5 each. One...

Seven thousand lottery tickets are sold for $\$ 5$ each. One ticket will win $\$ 2,000,$ two tickets will win $\$ 750$ each, and five tickets will win $\$ 100$ each. Let $X$ denote the net gain from the purchase of a randomly selected ticket. a. Construct the probability distribution of $X$. b. Compute the expected value $E(X)$ of $X .$ Interpret its meaning. c. Compute the standard deviation $\sigma$ of $X$.

Seven thousand lottery tickets are sold for $\$ 5$ each. One ticket will win $\$ 2,000,$ two tickets will win $\$ 750$ each,
and five tickets will win $\$ 100$ each. Let $X$ denote the net gain from the purchase of a randomly selected
ticket.
a. Construct the probability distribution of $X$.
b. Compute the expected value $E(X)$ of $X .$ Interpret its meaning.
c. Compute the standard deviation $\sigma$ of $X$.

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Key Concepts

Probability Distribution

A probability distribution is a mathematical function that gives the probabilities of occurrence of different possible outcomes in an experiment. It lists every potential outcome along with its probability, ensuring that the total probability sums to 1. This concept is fundamental in understanding how likely various results are when there is inherent randomness, such as in a lottery or any stochastic process.

Expected Value

The expected value is a measure of the central tendency of a random variable's probability distribution. It is computed as the sum of all possible values each multiplied by its probability. This value represents the average outcome if the experiment were repeated many times, helping to gauge the long-term benefit or cost of making a particular decision.

Standard Deviation

The standard deviation quantifies the amount of variation or dispersion in a set of outcomes. It is the square root of the variance, which is the average of the squared deviations from the expected value. A high standard deviation indicates that the outcomes are spread out over a wider range, while a low standard deviation suggests that they are clustered closely around the expected value.

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