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When we talk about Expected Value (EV), we are considering what you might gain or lose over many spins of the roulette. It gives you an average outcome, based on the probabilities. Let's take a closer look at how EV works in this exercise.
Bet 1 involves choosing a specific number, which is 23. If you win, you gain \(360 in total (\)350 plus your original \(10), but the chance of hitting that number is just \(\frac{1}{38}\). On the contrary, if any other number comes up, you lose your \)10. The EV for Bet 1 can be expressed as:
\[EV = \left(\frac{1}{38}\right) \times 360 + \left(\frac{37}{38}\right) \times (-10)\]
For Bet 2, you're wagering on the ball landing on a black number. The likelihood for this outcome is \(\frac{18}{38}\). Successful bets mean you earn \(20 in total (\)10 profit plus your \(10 back), or you lose your original \)10 if the ball doesn't land on black. Thus, the EV for Bet 2 is:
\[EV = \left(\frac{18}{38}\right) \times 20 + \left(\frac{20}{38}\right) \times (-10)\]
By examining these calculations, we can determine which bet statistically offers a better outcome over time by comparing their expected values.

Risk Assessment is determining what chance you are taking with a particular wager. Each bet offers different levels of risk and potential reward.
With Bet 1, there is a high risk because you're betting on a very unlikely event (only \(\frac{1}{38}\) probability of winning), but if it pays off, you earn a substantial amount (\(360). Essentially, you gamble on a single, more daring number, which offers high reward at a high risk.
On the other hand, Bet 2 is less risky due to the higher probability \(\frac{18}{38}\) that the ball will land on any of the black numbers. The reward here is smaller (a \)10 net gain), but so is the risk involved. This bet is more about incremental gains over a longer period compared to high stake scenarios like Bet 1.
Therefore, your assessment of each bet's risk can guide which wager might better suit your personal comfort levels and financial strategies. Some prefer the thrill of taking big chances with Bet 1, while others may lean toward steady wins with the lower risk of Bet 2.

Probability of Winning is simply the chance that a particular outcome will occur. Here, we look at how likely it is to win each bet based on the conditions given.
In Bet 1, the probability of landing on exactly the number 23 is \(\frac{1}{38}\). With only one winning number out of 38 possible outcomes on the roulette wheel, your chance of winning is pretty slim.
Conversely, Bet 2 offers a higher probability of winning. There are 18 black numbers, so the chances you pick any black number is \(\frac{18}{38}\). This means nearly half the time, you can expect to win if you bet on black.
Understanding these probabilities helps you make an informed decision. A higher probability sees a greater chance of frequent wins in the short term, whereas a lower probability might be riskier but could also provide higher rewards in a single lucky instance.