In a 6-deck Baccarat game, what's the probability of the banker winning at least 4 consecutive hands after the player wins 3 in a row?

[Huego213]

Step-by-Step Calculation:

1. Probability of Player Winning 3 Hands in a Row:
- In Baccarat, assuming no ties (which simplifies the calculation), the probability of the Player winning a single hand is approximately \( P(\text{Player wins}) = 0.4462 \).
- Therefore, the probability of the Player winning 3 hands in a row is:
\[
P(\text{Player wins 3 in a row}) = (0.4462)^3 \approx 0.0862
\]

2. Conditional Probability of Banker Winning at Least 4 Hands:
- After the Player has won 3 consecutive hands, we are interested in the probability of the Banker winning at least the next 4 hands.
- The probability of the Banker winning a single hand, given that the Player does not win (assuming no ties), is \( P(\text{Banker wins}) = 1 - P(\text{Player wins}) - P(\text{Tie}) \).
- Given the Banker's advantage, \( P(\text{Banker wins}) \) is approximately \( 0.4586 \).

 

[CR7]

3. Probability of Banker Winning at Least 4 Hands after Player's 3 Wins:
- Let's calculate the probability of the Banker winning exactly 4, 5, or 6 hands in a row after the Player has won 3 hands in a row.

- Probability of Banker Winning Exactly 4 Consecutive Hands:
\[
P(\text{Banker wins 4 in a row}) = (0.4586)^4 \approx 0.0872
\]

- Probability of Banker Winning Exactly 5 Consecutive Hands:
\[
P(\text{Banker wins 5 in a row}) = (0.4586)^5 \approx 0.0400
\]

- Probability of Banker Winning Exactly 6 Consecutive Hands:
\[
P(\text{Banker wins 6 in a row}) = (0.4586)^6 \approx 0.0184
\]

4. Probability of Banker Winning at Least 4 Consecutive Hands:
- To find the probability of the Banker winning at least 4 consecutive hands after the Player has won 3 hands in a row, we sum the probabilities of winning exactly 4, 5, or 6 hands in a row.
\[
P(\text{Banker wins at least 4 in a row}) = P(\text{Banker wins 4 in a row}) + P(\text{Banker wins 5 in a row}) + P(\text{Banker wins 6 in a row})
\]
\[
\approx 0.0872 + 0.0400 + 0.0184 \approx 0.1456
\]

Therefore, the probability of the Banker winning at least 4 consecutive hands after the Player wins 3 in a row in a 6-deck Baccarat game is approximately 0.1456 or 14.56%.

 

[swift]

I think probability of the banker winning any given hand in Baccarat is slightly greater than 50%, but the outcome of each hand is independent of the previous hands. Therefore, the probability of the banker winning at least 4 consecutive hands after the player wins 3 in a row

 

[Lazza]

After the player has won three straight, there is a 6.75% chance that the banker will win at least four hands in a row. This illustrates both the game of Baccarat's intrinsic randomness and the banker's tiny advantage.

 

[swift]

After the player has won three straight, there is a 6.75% chance that the banker will win at least four hands in a row. This illustrates both the game of Baccarat's intrinsic randomness and the banker's tiny advantage.

i think Despite the small advantage, it is still possible for a player to have a winning streak and for the banker to have a losing streak. This is because each hand is independent of the previous hand, and the outcome is determined by chance. The probability of the banker winning four hands in a row after the player has won three straight is low, but it is not impossible.