Suppose you draw a card from an ordinary deck of playing cards. Find each of the following odds. a. What are the odds in favor of drawing a red king? b. What are the odds against drawing a red king? a. What are the odds in favor of drawing a red king? \( \square \) to \( \square \) (Simplify your answers.) b. What are the odds against drawing a red king? \( \square \) to \( \square \) (Simplify your answers.)

Let's solve the problem step by step. ### Step 1. Determine the Total Number of Cards An ordinary deck has \(52\) cards. ### Step 2. Identify the Favorable Outcomes A "red king" is a king that is either of hearts or diamonds. There are exactly \(2\) such cards: - King of Hearts - King of Diamonds So, the number of favorable outcomes (drawing a red king) is: \[ \text{Favorable outcomes} = 2 \] ### Step 3. Identify the Unfavorable Outcomes The unfavorable outcomes are the cards that are not a red king. That is: \[ \text{Unfavorable outcomes} = 52 - 2 = 50 \] ### Step 4. Calculate the Odds #### a. Odds in Favor of Drawing a Red King The odds in favor are given by the ratio of the number of favorable outcomes to the number of unfavorable outcomes: \[ \text{Odds in favor} = 2 \text{ to } 50 \] We simplify this ratio by dividing both numbers by their greatest common divisor, which is \(2\): \[ \frac{2 \div 2}{50 \div 2} = \frac{1}{25} \] Thus, the odds in favor are: \[ 1 \text{ to } 25 \] #### b. Odds Against Drawing a Red King The odds against are the reverse ratio: unfavorable outcomes to favorable outcomes: \[ \text{Odds against} = 50 \text{ to } 2 \] Simplify by dividing both numbers by \(2\): \[ \frac{50 \div 2}{2 \div 2} = \frac{25}{1} \] So, the odds against are: \[ 25 \text{ to } 1 \] ### Final Answers - **a. Odds in favor of drawing a red king:** \(1\) to \(25\) - **b. Odds against drawing a red king:** \(25\) to \(1\)