Question

A card is drawn from a pack of $52$ cards. A gambler bets that it is a spade or an ace. What are the odds against his winning this bet

• A. $17:52$
• B. $52:17$
• C. $9:4$
• D. $4:9$

Answer 1

The correct option is C

$9:4$

Explanation for the correct option:

Let events $\alpha$: a spade is drawn and event $\beta$ an ace is drawn.

then, The probability of winning the bet $=P(\alpha or\beta )$

$=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}$ $\left[\mathbf{\because }\mathit{P}\mathbf{(}\mathit{A}\mathbf{}\mathit{o}\mathit{r}\mathbf{}\mathit{B}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{P}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{+}\mathit{P}\mathbf{\left(}\mathit{B}\mathbf{\right)}\mathbf{-}\mathit{P}\mathbf{(}\mathit{A}\mathbf{\cap }\mathit{B}\mathbf{)}\right]$

$$\begin{gathered} =\frac{16}{52} \\ =\frac{4}{13} \end{gathered}$$

∴ Probability of losing the bet $\begin{array}{rcl}& =& 1-\frac{4}{13}\end{array}$

$\begin{array}{rcl}& =& \frac{9}{13}\end{array}$

Thus, the odds against winning the bet $=\frac{9}{13}:\frac{4}{13}$

$=\mathbf{9}\mathbf{:}\mathbf{4}$

Hence, Option ‘C’ is Correct.