Final answer:
The value of a $1 state lottery ticket where five numbers are chosen from 1 to 30 can be computed by determining the expected value, which involves the probabilities of winning the grand prize ($100,000) or a secondary prize ($100) and the loss of the ticket price ($1).
Step-by-step explanation:
In a state lottery where five numbers are randomly chosen from the numbers 1 to 30, the value of a $1 lottery ticket can be calculated by determining the expected value of the ticket based on the probabilities and payouts of winning.
For the grand prize, you must choose all 5 numbers correctly. The probability of doing this is 1 divided by the number of combinations possible (from 30 choose 5). Assuming you pick all 5 numbers correctly, your profit is $100,000.
For the secondary prize, you must pick 4 of the 5 numbers correctly. This involves calculating the number of combinations with 4 correct numbers and one incorrect number and its associated probability. Your profit for this scenario is $100.
If you do not pick either 4 or all 5 numbers correctly, you do not win any prize and your loss is the cost of the ticket: $1.
By multiplying the profit by the probability of each scenario and adding them together, you can find the expected value of the lottery ticket. If the expected value is positive, the ticket is valuable; if it is negative, it means a loss is expected over the long term. The ticket's value is essentially its expected gain or loss.