Final answer:
When calculating the expected average winnings per raffle for purchasing four tickets, you need to consider the probability of winning for each option, with or without replacement, and calculate the expected value based on these odds and the prize values.
Step-by-step explanation:
If the awards are for attendance, participation, and other factors, and you can win more than once, this introduces the concept of probability with or without replacement. To calculate your expected average winnings per raffle when you always purchase four tickets, you would need to know the odds of winning for each ticket, the total number of tickets in play, and if tickets are removed after winning (without replacement) or not (with replacement). If the probabilities change each time because tickets are removed, this is without replacement; if the same winning chances are maintained each time regardless of previous results, this is with replacement.
Understanding these concepts is crucial, as they affect the calculation of expected value. The expected value is determined by multiplying the probability of each outcome by the value of that outcome and then summing all these products. This gives you the average amount you could expect to win per ticket over the long run, which you would then multiply by the number of tickets you purchase (four, in this case). However, if each ticket can only win once (without replacement), the calculations would become more complex as the odds change with each draw.
On the subject of incentive models impacting academic performance, the situation described suggests that removing incentives might lower overall student achievement. This is a different area of study, typically considered within educational psychology or behavioural economics, and is separate from the mathematical concept of expected value.