For an advertising promotion, an auto dealer hands out 1000...

For an advertising promotion, an auto dealer hands out 1000 lottery tickets with a prize of a new car worth $$\$ 25,000 .$$ For someone with a single ticket, what is the standard deviation for the amount won? (A) $$\$ 7.07$$ (B) $$\$ 25.00$$ (C) $$\$ 49.95$$ (D) $$\$ 790.17$$ (E) $$\$ 624,375$$

For an advertising promotion, an auto dealer hands out 1000 lottery tickets with a prize of a new car worth $$\$ 25,000 .$$ For someone with a single ticket, what is the standard deviation for the amount won?
(A) $$\$ 7.07$$
(B) $$\$ 25.00$$
(C) $$\$ 49.95$$
(D) $$\$ 790.17$$
(E) $$\$ 624,375$$

Key Concepts

Variance

Variance measures the spread of a random variable's possible values by calculating the average of the squared differences from the expected value. It quantifies how much the outcomes differ from the mean, offering a sense of the variability present in the distribution. In gambling or lottery contexts, variance helps assess the risk associated with a random outcome.

Discrete Random Variables

In probability theory, a discrete random variable is a variable that can take on a countable number of distinct values. Each value is associated with a probability, and the sum of all probabilities is 1. This concept is used to model outcomes of experiments where there are a finite number of results, such as winning a prize or winning nothing in a lottery.

Bernoulli Distribution

A Bernoulli distribution is a special case of a discrete probability distribution where there are only two possible outcomes, typically termed 'success' and 'failure'. This distribution is characterized by a single parameter representing the probability of success. In scenarios like a lottery ticket that either wins a prize or does not, the Bernoulli model is an ideal framework.

Expected Value

The expected value of a random variable represents the long-run average outcome of a random event after many repetitions. It is calculated by summing the products of each outcome and its corresponding probability. In problems involving winnings or losses, the expected value gives an indication of the average amount one would win per trial over many trials.

Standard Deviation

The standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the original data. It indicates, on average, how much the outcomes differ from the expected value. In contexts like lottery winnings, the standard deviation offers insight into the volatility or uncertainty of the prize amount.

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