The Unfair Coin Probability Calculator is designed to compute the probability of various outcomes when flipping a biased coin multiple times. Unlike a fair coin, which has equal chances of landing heads or tails, an unfair coin has skewed probabilities, making certain outcomes more likely. This calculator helps users understand and quantify the likelihood of specific sequences of results over a series of flips, providing valuable insights for statistical analysis, game theory, and decision-making processes.
Formula of Unfair Coin Probability Calculator
The fundamental formula used by the Unfair Coin Probability Calculator is:
P(Outcome) = P(H)^n_H * P(T)^n_T
Where:
P(Outcome)is the probability of the specific outcome.P(H)is the probability of getting heads.P(T)is the probability of getting tails.n_His the number of times you expect to get heads.n_Tis the number of times you expect to get tails.
For instance, to calculate the probability of getting heads three times and tails twice in five flips of an unfair coin, with the probability of heads being 0.6 and the probability of tails being 0.4, the formula simplifies to:
P(3H, 2T) = 0.6^3 * 0.4^2
This formula is central to the calculator's functionality. Enabling users to input their specific probabilities and expected outcomes to receive accurate results.
Pre-calculated Table for Common Searches
| Scenario | Probability of Heads (P(H)) | Probability of Tails (P(T)) | Outcome | Probability Formula | Example Probability |
|---|---|---|---|---|---|
| 1 | 0.6 | 0.4 | 3H, 2T | P(Outcome) = 0.6^3 * 0.4^2 | 0.3456 |
| 2 | 0.7 | 0.3 | 4H, 1T | P(Outcome) = 0.7^4 * 0.3^1 | 0.2401 |
| 3 | 0.5 | 0.5 | 2H, 3T | P(Outcome) = 0.5^2 * 0.5^3 | 0.3125 |
| 4 | 0.8 | 0.2 | 5H, 0T | P(Outcome) = 0.8^5 * 0.2^0 | 0.32768 |
| 5 | 0.4 | 0.6 | 1H, 4T | P(Outcome) = 0.4^1 * 0.6^4 | 0.2592 |
This table offers a snapshot of how different probabilities of heads and tails affect the overall outcome of multiple coin flips.
Example of Unfair Coin Probability Calculator
Consider a scenario where you have an unfair coin with a 70% chance of landing heads (P(H) = 0.7) and a 30% chance of landing tails (P(T) = 0.3). If you want to know the probability of flipping this coin five times and getting exactly 3 heads and 2 tails. The calculation would be:
P(3H, 2T) = 0.7^3 * 0.3^2
By computing this, you receive a precise probability, showcasing the calculator's ability to handle varied and complex scenarios.
Most Common FAQs
A1: Yes, by setting the probability of heads and tails to 0.5 each. The calculator can model scenarios involving fair coins.
A2: While the basic formula focuses on the number of heads and tails. More complex sequences may require additional calculations or the use of combinatorics to consider order.