Question

Two cards are drawn at random from a pack of playing cards. If the odds against that one card is a heart and the other is an ace, are $m:1$, find m.

Answer 1

Answer: 25

Heart and not Ace = 12
Ace and not heart = 3
Ace and heart = 1
E : one card is heart and other Ace
$P(E)=\frac{{}^{12}{C}_1\cdot {}^3{C}_1+{}^3{C}_1\cdot {}^1{C}_1+{}^{12}{C}_1\cdot {}^1{C}_1}{{}^{52}{C}_2}=\frac{(36+3+12)2}{52.51}=\frac{51.2}{52.51}=\frac{1}{26}$
$\Rightarrow \text{ odds in favour }1:25$
$\therefore \text{ odds against }25:1\Rightarrow 25$