A card is drawn from a deck of 52 playing cards.
a) Find the odds against drawing a red eight or a black eight.
b) Find the odds against drawing a red eight and a black eight.
a) The odds against drawing a red eight or a black eight, in lowest terms, are $\square: \square$.
(Type whole numbers.)
b) The odds against drawing a red eight and a black eight, in lowest terms, are $\square: \square$.
(Type whole numbers.)
Answer
The odds against drawing a red eight and a black eight are not applicable in this case, because you can't draw two cards at once from a single deck. The question would make sense if we were drawing two cards in succession, but as it stands, the odds are undefined.
Step 1 ：In a standard deck of 52 playing cards, there are 2 red eights (one heart and one diamond) and 2 black eights (one spade and one club). So, the total number of favorable outcomes (drawing a red eight or a black eight) is \(2 + 2 = 4\).
Step 2 ：The total number of possible outcomes is 52 (the total number of cards in the deck).
Step 3 ：The odds against an event are calculated as (number of unfavorable outcomes) : (number of favorable outcomes).
Step 4 ：The number of unfavorable outcomes is the total number of outcomes minus the number of favorable outcomes, which is \(52 - 4 = 48\).
Step 5 ：So, the odds against drawing a red eight or a black eight are \(48 : 4\).
Step 6 ：We can simplify this ratio by dividing both sides by 4, to get \(\boxed{12 : 1}\).
Step 7 ：The odds against drawing a red eight and a black eight are not applicable in this case, because you can't draw two cards at once from a single deck. The question would make sense if we were drawing two cards in succession, but as it stands, the odds are undefined.
