A lottery ticket has a grand prize of $28 million. The probability of winning the grand prize is .000000023. Based on the expected value of the lottery ticket, would you pay $1 for a ticket? Show your calculations and reasoning clearly.

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To determine if it is worth paying $1 for a ticket, we need to calculate the expected value.

Expected value = (Probability of winning grand prize) × (Amount of grand prize) - (Ticket cost)

Given:
Probability of winning grand prize = 0.000000023
Amount of grand prize = $28 million or $28,000,000
Ticket cost = $1

Expected value = (0.000000023) × ($28,000,000) - ($1)
Expected value = $0.644 - $1
Expected value = -$0.356

The expected value of the lottery ticket is -$0.356. This means that on average, each ticket will cost you $0.356, resulting in a loss. Therefore, it does not make financial sense to pay $1 for a ticket, as you can expect to lose money in the long run.

To determine whether it's worth paying $1 for a lottery ticket with a grand prize of $28 million and a winning probability of .000000023, we can calculate the expected value of the ticket.

The expected value is calculated by multiplying the value of each outcome by its probability of occurring, then summing up these values. In this case, we have two possible outcomes: winning the grand prize or not winning anything.

1. Winning the grand prize:
The value of this outcome is $28 million.
The probability of this outcome is .000000023.

2. Not winning anything:
The value of this outcome is $0.
The probability of this outcome is 1 - .000000023, which is approximately 1.

Now let's calculate the expected value:

Expected Value = (Value of winning outcome * Probability of winning) + (Value of losing outcome * Probability of losing)
= ($28,000,000 * .000000023) + ($0 * 1)
= $0.644

The expected value of the lottery ticket is $0.644.

Typically, if the expected value is positive, it would be worth paying for the ticket, as it suggests a net gain over multiple trials. If the expected value is negative, it would not be worth purchasing.

In this case, since the expected value is positive ($0.644), it would be worth paying $1 for the ticket from a purely mathematical standpoint. However, it's important to note that the expected value represents long-term average outcomes, and in reality, you may or may not win the grand prize. Therefore, personal preferences, budget constraints, and the understanding that winning is extremely unlikely should also be considered before deciding whether to purchase a ticket.