Questions: One card is randomly selected from a deck of cards. Find the odds against drawing an eight. The odds against drawing an eight are 12 . 1 . (Simplify your answers.)

Solution Steps

We need to find the odds against drawing an eight from a standard deck of 52 playing cards. A standard deck has 4 eights (one for each suit: hearts, diamonds, clubs, and spades).

The probability of drawing an eight is given by the ratio of the number of favorable outcomes to the total number of outcomes. There are 4 eights in a deck of 52 cards.

\[ P(\text{drawing an eight}) = \frac{4}{52} = \frac{1}{13} \]

The odds against an event are calculated as the ratio of the probability of the event not occurring to the probability of the event occurring.

The probability of not drawing an eight is:

\[ P(\text{not drawing an eight}) = 1 - P(\text{drawing an eight}) = 1 - \frac{1}{13} = \frac{12}{13} \]

Thus, the odds against drawing an eight are:

\[ \text{Odds against drawing an eight} = \frac{P(\text{not drawing an eight})}{P(\text{drawing an eight})} = \frac{\frac{12}{13}}{\frac{1}{13}} = 12:1 \]

Final Answer