Final answer:
The odds against drawing a spade greater than two and less than nine from a standard deck of 52 cards are calculated by subtracting the number of favorable spades from the total number of cards, which gives us 46:6. The simplified odds ratio is 23:3.
Step-by-step explanation:
The student is asking to find the odds against drawing a spade that is greater than two and less than nine from a standard deck of 52 cards. In each suit of a standard deck, there are 13 cards, and if we consider the spades suit, the cards that are greater than two and less than nine are 3, 4, 5, 6, 7, and 8, totaling 6 cards.
To calculate the odds against drawing one of these spades, we find the number of unfavorable outcomes, which is the total number of cards minus the number of favorable spades (52 - 6 = 46), and then we express it as a ratio against the number of favorable outcomes (6).
The odds against drawing a spade that is greater than two and less than nine are, therefore, 46:6, which can be simplified by dividing both numbers by their greatest common divisor. In this case, the simplified odds ratio is 23:3.