You have a bankroll of \$2275 at a blackjack table. The croupier draws a single card from a standard 52-card deck, and you may bet on the event that the card is a face card or an ace. The croupier offers you decimal odds of 4.50 on this bet. Using the Kelly criterion, how many dollars of your \$2275 bankroll should you stake to maximise your long-run growth rate?

Let's solve this in three moves: first the true chance of winning, then the net odds the croupier is really paying, and finally the Kelly bet size that ties them together. Step 1 — the true probability.A single card is drawn from a standard 52-card deck, so there are 52 equally likely outcomes. The winning ranks are Jack, Queen, King and Ace — 4 ranks across 4 suits, which is $4 \times 4 = 16$ cards — so the chance of winning is \begin{equation} p = \frac{16}{52} = \frac{4}{13} \approx 0.3077, \end{equation} and the chance of losing is \begin{equation} q = 1 - \frac{4}{13} = \frac{9}{13} \approx 0.6923. \end{equation} Step 2 — turn the odds into the net odds $b$.The Kelly formula needs the net odds $b$, meaning the profit you keep per \$1 staked when you win (you also get your \$1 stake back), so our first job is to turn the quoted odds into $b$. Decimal odds $o$ quote the total returned per \$1 staked, stake included, so we subtract that returned \$1 to leave the profit: \begin{equation} b = o - 1 = 4.5 - 1 = 3.5. \end{equation} It is also worth seeing what probability those odds imply, because that is what tells you the bet is worth making. A betting line is a probability in disguise: the break-even probability behind decimal odds $o$ is $1/o = 1/4.5 = \frac{2}{9}$. The croupier is pricing the bet as if your chance were about 22.2%, but you have worked out the true chance is about 30.8%. That gap is your edge, and it is exactly why the bet is profitable (for reading odds as probabilities, see the Prediction Markets: Odds, Implied Probability, and Arbitrage lesson). Step 3 — the Kelly criterion.When you have an edge and you will bet repeatedly, the question is what fraction of your bankroll to stake. Bet too little and your money grows too slowly; bet too much and a single losing streak wipes you out. The Kelly criterion picks the fraction that maximises your long-run growth rate. For a simple win-or-lose bet paying net odds $b$, with win probability $p$ and loss probability $q = 1 - p$, the optimal fraction of your bankroll is \begin{equation} f^{*} = \frac{bp - q}{b}. \end{equation} The numerator $bp - q$ is your expected profit per \$1 staked (you gain $b$ with probability $p$ and lose $1$ with probability $q$); dividing by $b$ rescales that edge into the right fraction to wager. If $f^{*}$ comes out zero or negative you have no edge and should not bet at all (this is the formula derived in the The Kelly Criterion: A Guide to Smarter Betting lesson). Step 4 — plug in the numbers.With $b = 3.5$, $p = \frac{4}{13}$ and $q = \frac{9}{13}$, \begin{equation} f^{*} = \frac{3.5 \times \frac{4}{13} - \frac{9}{13}}{3.5} = \frac{\frac{5}{13}}{3.5} = \frac{10}{91}. \end{equation} So Kelly tells you to stake about 11.0% of your bankroll, and on \$2275 that is \begin{equation} \frac{10}{91} \times 2275 = 250. \end{equation} So you should bet \$250 on this draw.

Decimal Betting - Cards #2 | Betting And Sizing

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