SOLUTION: What is the probability of betting on all odds and winning? What is the probability of betting on the 1st dozen and winning? A player bets on the number 4 in the first game

Question 1196889: What is the probability of betting on all odds and winning?
What is the probability of betting on the 1st dozen and winning?
A player bets on the number 4 in the first game; then bets on number 28 in the second game. Did the player’s odds of winning change from the first to the second game? Explain.
What is a better bet: all reds or all evens? Explain. (Hint: 0 and 00 are not included in an all even bet)
If a player always bets on 00, what is the expected winnings (or loses) when playing 20 games of Roulette given the player receives $100 if they win and owes $1 if they lose?
Answer by ElectricPavlov(122) (Show Source): You can put this solution on YOUR website!
**1. Probability of Betting on All Odds and Winning**
* **American Roulette:** There are 18 odd numbers, 18 even numbers, 0, and 00.
* Probability of winning on an odd number: 18/38 = 9/19
* **European Roulette:** There are 18 odd numbers, 18 even numbers, and 0.
* Probability of winning on an odd number: 18/37
**2. Probability of Betting on the 1st Dozen and Winning**
* There are 12 numbers in the first dozen (1-12) out of 38 total numbers (American) or 37 total numbers (European).
* Probability of winning: 12/38 (American) or 12/37 (European)
**3. Player's Odds of Winning on 4 vs. 28**
* The player's odds of winning do **not** change between betting on 4 and 28.
* Each individual number on the roulette wheel has an equal probability of occurring on each spin.
**4. Better Bet: All Reds or All Evens**
* **In theory:** Both bets have a very slightly higher probability of losing due to the presence of 0 and 00 on the American roulette wheel.
* **Practical Considerations:** There might be slight biases in the wheel or ball that could favor one over the other in the long run. However, these biases are usually minimal and difficult to predict.
**5. Expected Winnings/Losses Betting on 00 for 20 Games**
* **Probability of winning on 00:** 1/38 (American) or 1/37 (European)
* **Probability of losing on 00:** 37/38 (American) or 36/37 (European)
* **Expected Winnings (American):**
* (1/38) * $100 + (37/38) * (-$1) = $2.63 - $0.97 = -$0.34
* **Expected Winnings (European):**
* (1/37) * $100 + (36/37) * (-$1) = $2.70 - $0.97 = -$0.27
* **Expected Winnings (20 Games):**
* (Expected Winnings per Game) * Number of Games
* American: -$0.34 * 20 = -$6.80
* European: -$0.27 * 20 = -$5.40
**Therefore, when betting on 00 for 20 games:**
* **American Roulette:** You can expect to lose approximately $6.80.
* **European Roulette:** You can expect to lose approximately $5.40.
**Important Note:**
* These calculations are based on the mathematical probabilities. In reality, actual outcomes may vary due to the random nature of the game.
* Roulette is a casino game with a house edge, meaning the casino has a built-in advantage over the player in the long run.

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