Final answer:
To maximize profits, the firm should balance the expected profits from selling T-shirts at full price against the potential losses from unsold inventory post-event. Without complete information, a precise number can't be provided, but typically, the firm should avoid exceeding the demand level with a high cumulative probability and not under-produce compared to the demand with a steep increase in cumulative probability.
Step-by-step explanation:
To determine how many T-shirts the silk-screening firm should produce to maximize profits for the Bolder Boulder event, we need to examine the expected demand distribution and the profit or loss associated with each potential level of production. The firm makes a profit of $6.00 per T-shirt when sold at $12.00 considering a cost of $6.00 to produce each one. However, if T-shirts are sold post-event at $3.00, the firm incurs a loss of $3.00 per unsold T-shirt.
We should calculate the expected profit for each production level by multiplying the profit or loss per T-shirt by the probability of demand at each level. However, this information is not provided in complete extent. Normally, we would create a payoff table with all possible profits for the respective levels of demand. Then, we would identify which production quantity has the highest expected profit, factoring in the probability of each demand level occurring.
Without complete information, we cannot provide a numeric answer. But, in general, the firm should not produce more T-shirts than the expected demand where the cumulative probability is close to 1, and should avoid producing significantly less than the demand level where the cumulative probability starts to increase steeply, as this represents a strong chance of selling a large quantity at full price.