A card is drawn and replaced in an ordinary pack of 52 playing cards. Minimum number of times must a card be drawn so that there is at least an even chance of drawing a heart?
What does the second sentence of the question mean?
Why should there be any change in the chance of drawing a heart, as the cards are being "replaced" (kept back into the pack after drawing)?
Best Answer
For one draw, we have a chance $\frac{1}{4}$ of drawing a heart. For $n$ cards, the chance we don't draw any hearts at all is $(\frac{3}{4})^n$ as we need (independent, as we put cards back) events of not drawing a heart. So the complementary probability $1-(\frac{3}{4})^n$ computes the probability of drawing at least a heart, which is what was asked.
So for what $n$ (number of draws) does this become $\ge \frac{1}{2}$?
Solve for $n$ using logarithms, say.