Question

Six playing cards are lying face down on a table, two of them are kings. Two cards are drawn at random. Let \(a\) denote the probability that at least one of the cards drawn is a king, and \(b\) denote the probability of not drawing a king. The ratio \(a/b\) is

• A. \(\ge 0.25 \text{ and }<0.5\)
• B. \(\ge 0.5 \text{ and }<0.75\)
• C. \(\ge 0.75 \text{ and }<1.0\)
• D. \(\ge 1.0 \text{ and }<1.25\)
• E. \(\ge 1.25\)

Hint:

Use complements for “at least one” events: \(P(\textat least one king)=1-P(\textno king)\). It’s faster and avoids casework.

Answer 1

Answer: (E)

Step 1: There are \(2\) kings and \(4\) non-kings among \(6\) cards. Two cards are drawn without replacement. Total outcomes \(=\binom{6}{2}=15\).
Probability of no king (both non-kings): \[ b=\frac{\binom{4}{2}}{\binom{6}{2}}=\frac{6}{15}=\frac{2}{5}=0.4. \] Step 2: Probability of at least one king: \[ a=1-b=1-\frac{2}{5}=\frac{3}{5}=0.6. \] Step 3: Ratio \[ \frac{a}{b}=\frac{\frac{3}{5}}{\frac{2}{5}}=\frac{3}{2}=1.5\ (\ge 1.25). \] Hence the interval is \(\boxed{\text{(E) }\ge 1.25}\).