Expected Lotto winnings. The chance of winning a famous...

Expected Lotto winnings. The chance of winning a famous lottery game is 1 in approximately 50 million. Suppose you buy a $\$ 1$ lottery ticket in anticipation of winning the $\$ 7$ million grand prize. Calculate your expected net winnings for this single ticket. Interpret the result.

Key Concepts

Expected Value

The expected value is a fundamental concept in probability and statistics that represents the weighted average of all possible outcomes. It is calculated by multiplying each outcome by its corresponding probability and summing these products. This concept is especially useful in understanding the fairness or potential profitability of bets and investments over the long run.

Net Winnings

Net winnings refer to the profit or loss that remains after deducting any costs associated with the transaction from the gross return. In the context of the lottery, this means subtracting the price of the ticket from the prize if you win, and considering the loss of the ticket cost if you do not win.

Probability

Probability measures the likelihood of a particular event occurring. In the lottery example, the extremely low probability of winning (1 in 50 million) is a key factor in determining the expected value, as it significantly diminishes the impact of the large prize on the overall calculation.

Interpretation of Expected Value

The interpretation of the expected value in this scenario is that although there is a chance to win a large sum, the probability is so low that the average outcome over many plays is a loss. A negative expected net winning means that for each ticket purchased, you expect to lose money in the long run, indicating that the lottery is not a favorable bet.

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