Discover a Gambling Game That Seems Fair but Always Pays Out!

Ready to hear about a game that, if it existed, you’d totally want to join? It might seem all fair at first glance, but here’s the twist – it’s a game where you’re always bound to make some cash.

The Rules

Here’s how this always-win game rolls:

The dealer keeps flipping a coin. Before each flip, you, the player, place your bet on either heads or tails. Let’s say you’re feeling the heads vibe. If heads shows up on the first flip, you pocket 2 cents and you’re out. But if tails pops up? No worries, they flip again. If heads appears on this next toss, you score 4 cents and you’re done. Tails again? Another flip it is. When heads finally makes its debut, you snag 8 cents. This continues until your bet (either heads or tails) finally shows. If it takes n flips for your choice to come through, you’ll be cashing in \(2^n\) cents. Sounds fun, right?

How Much Would You Pay to Play?

Put yourself in the player’s shoes for a sec. What’s the max you’d fork out as an entry fee? At the very least, you’re getting 2 cents back. So if you’re paying a 2-cent entry fee, you’re breaking even. And with a 50% shot at 4 cents, 3 cents for entry doesn’t sound too shabby. But here’s the kicker – whatever the entry fee, the math’s on your side!

Expected Value of the Game

Let’s get our nerd on and talk math. Label the number of flips as \(n\) and the expected value as \(K\). On the nth flip, you’re looking at a potential \(2^n\) cents win. Given there’s a 1 in 2 chance for each flip, our expected value \(K\) can be worked out as:

\(\displaystyle K=({2^1}×\frac{1}{2^1})+({2^2}×\frac{1}{2^2})+({2^3}×\frac{1}{2^3})+\)
\(\displaystyle ・・・+({2^n}×\frac{1}{2^n})\)
\( \displaystyle K=\sum_{n=1}^{m}{({2^n}×\frac{1}{2^n})} \)

And since the sky’s the limit with how many times you can play, let \(m\) soar to infinity. Plugging this into our previous equation:

\( K=1+1+1+1+1+1+・・・＝∞ \)

There you have it! Our expected value K is looking pretty infinite. Meaning, the more rounds you play, the richer you could get. Just a heads-up though: scoring a cool 100,000 cents is a 1 in 100,000 shot. So if you’re diving in, make sure you’ve got some funds to back you up!