Answer:
The optimal strategy for Bob is buying for shine (unless he can watch a forecast to know the next day weather).
Explanation:
This is a typical problem of hopes to win vs hopes to lose. Let's analyze each of the strategies Bob can adopt in both kinds of weather.
Bob buy for rain:
Bob will buy 500 umbrellas for a cost of $5 each. This is a total cost of $2500.
If it rain, Bob can sell all umbrellas for $10 each. This gives a maximum revenue of $5000. Therefore the maximum profit is $2500. Remember that:
Profit= Revenue - Cost
If it's a sunny day, Bob can only sell 100 umbrellas for $10 each. This gives a maximum revenue of $1000. Therefore the maximum profit is -$1500. That means that in this case, the minimum loss is $1500.
Bob buy for Shine:
Bob will buy 100 umbrellas for a cost of $5 each and 1000 sunglasses for a cost of $2 each. This is a total cost of $2500.
If it's a sunny day, Bob can only sell all umbrellas for $10 each and all sunglasses for $5. This gives a maximum revenue of $6000. Therefore the maximum profit is $3500.
If it rains, Bob can sell only sell all the 100 umbrellas for $10 each but none of the sunglasses. Therefore the maximum profit is $1000. Therefore the maximum profit is -$1500. That means that in this case, the minimum loss is $1500.
In both cases, the worst-case scenario is the same: a loss of $1500.
Nevertheless in the best case scenario buying to shine gives a bigger profit. Therefore if the risk is the same, is better to go for the strategy with better profits.