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Ask The Wizard #360
In the film Casino Royale, during the dramatic conclusion of a poker tournament, the four contestants reveal their cards:
- Flush
- Full house
- A full house (with a different set of values than the first one)
- Straight flush
What is the probability of that?
I needed to conduct a simulation for this scenario. My simulation assumed that no player folds their hand. After simulating nearly 2.2 billion rounds, this outcome occurred 312 times. This results in a probability close to 1 in seven million.
In the Roll to Win craps games, participants can place both lay and place-to-lose wagers. Here are the odds associated with the place-to-lose bets:
- 4 and 10: 5 to 11
- 5 and 9: 5 to 8
- 6 and 8: 4 to 5
The lay bets provide fair odds; however, there is a 5% commission taken from the winning amount when a player wins.
My inquiry is about which betting option provides superior odds?
The following chart illustrates the house edge for both types of bets according to the number wagered. It can be observed that the house edge is generally lower for lay bets, except for points 6 and 8.
House Edge Analysis for Place to Lose and Lay Bets
| Number | Place to Lose | Lay |
|---|---|---|
| 4 or 10 | 3.03% | 1.67% |
| 5 or 9 | 2.50% | 2.00% |
| 6 or 8 | 1.82% | 2.27% |
We received the upcoming question courtesy of the Riddler Express .
Let's consider NFL regulations and analyze this scenario:
- The Red team is trailing by 14 points towards the end of the match.
- The Red team will have two more opportunities with the ball.
- The Blue team will not have any further possessions.
- For simplicity, let's disregard field goals and safeties, since the Red team needs to score two touchdowns to have a chance of victory.
- If the game enters overtime, both teams would have an equal 50% chance of winning. A tie is not a possible outcome.
- The success rate for a one-point conversion after a touchdown is guaranteed at 100%.
- The likelihood of successfully executing a two-point conversion is represented by p.
At what value of p should the Red team have no preference between attempting a two-point conversion and opting for the kick following their initial touchdown (currently 8 points behind)?
Let’s define p as the point of indifference between pursuing the two-point conversion versus taking the kick.
If the first attempt for a two-point conversion is successful, the Red team has the option to attempt a kick afterward and win the game.
If the initial attempt for a two-point conversion fails, the Red team must try for a conversion again after scoring a second touchdown and then aim to win during overtime.
The probability of scoring when going for a two-point conversion after the first touchdown is expressed as p + (1-p)*p/2. We will set this equal to the 50% chance of winning by making the kick after the initial touchdown and solve for p.
p + (1-p)*p/2 = 1/2
2p + (1-p)*p = 1
3p - p^2 = 1
p^2 - 3p + 1 = 0
Applying the quadratic formula allows us to determine p:
p = (3 +/- sqrt(5))/2
Selecting the negative solution ensures p remains within the range of 0 to 1, yielding p = (3-sqrt(2))/2, approximately equal to 0.381966011250105.
This topic is currently being discussed in my forum located at Wizard of Vegas .