Fair vs. Biased Coin
You have two coins. Coin A is a fair coin (P(Heads) = 0.5). Coin B is a biased coin (P(Heads) = 0.7). You randomly pick one of the coins (with equal probability) and flip it twice. You observe two Heads (HH).
What is the probability that you picked the biased coin (Coin B)?
Related Concepts
Hint
- Define your hypotheses: HF = Fair coin picked, HB = Biased coin picked.
- What are the prior probabilities P(HF) and P(HB)?
- Define the evidence E = observing two Heads (HH).
- Calculate the likelihood of E under each hypothesis:
- P(E | HF): If fair, P(HH) = P(H) × P(H) = ?
- P(E | HB): If biased, P(HH) = P(H) × P(H) = ?
- Calculate the total probability of the evidence P(E) using:
P(E) = P(E | HF)P(HF) + P(E | HB)P(HB) - Apply Bayes' Theorem to find P(HB | E).
Explanation: Fair vs. Biased Coin
You have two coins: one is normal (fair, 50% Heads), the other is tricky (biased, 70% Heads). You pick one by chance (50/50 which one you get) and flip it twice. Both times, it comes up Heads!
Now, what's the chance you actually picked the biased coin?
- Initially, there was a 50% chance you picked the biased coin.
- Getting two Heads is more likely if you have the biased coin (70% Heads) than if you have the fair coin (50% Heads).
- So, the evidence (two Heads) should make us increase our belief that we picked the biased coin. Bayes' Theorem helps us figure out exactly how much.
We will use Bayes' Theorem to update our belief about which coin was picked, given the evidence of two heads.
Let the events/hypotheses be:
- F: The Fair coin was picked.
- B: The Biased coin was picked.
- E: The evidence observed is two Heads (HH) in two flips.
We want to find P(B | E).
1. Define Prior Probabilities
Since a coin is picked randomly with equal probability:
P(F) = 0.5(Prior probability of picking the Fair coin)P(B) = 0.5(Prior probability of picking the Biased coin)
2. Calculate the Likelihood of the Evidence under each Hypothesis
The two flips are independent given the type of coin.
- Likelihood of two Heads if the Fair coin was picked (P(H|F) = 0.5):
P(E | F) = P(Head on 1st flip | F) × P(Head on 2nd flip | F)
P(E | F) = 0.5 × 0.5 = (0.5)2 = 0.25 - Likelihood of two Heads if the Biased coin was picked (P(H|B) = 0.7):
P(E | B) = P(Head on 1st flip | B) × P(Head on 2nd flip | B)
P(E | B) = 0.7 × 0.7 = (0.7)2 = 0.49
3. Calculate the Total Probability of the Evidence (P(E))
Using the Law of Total Probability:
P(E) = P(E | F) × P(F) + P(E | B) × P(B)
P(E) = (0.25 × 0.5) + (0.49 × 0.5)
P(E) = 0.125 + 0.245
P(E) = 0.370
This is the overall probability of observing two heads, considering either coin could have been picked.
4. Apply Bayes' Theorem to Find P(B | E)
Bayes' Theorem states:
P(B | E) = [P(E | B) × P(B)] / P(E)
P(B | E) = (0.49 × 0.5) / 0.370
P(B | E) = 0.245 / 0.370
P(B | E) ≈ 0.66216216...
Final Result
The probability that you picked the biased coin, given that you observed two heads, is:
P(Biased Coin | Two Heads) ≈ 0.662
Or about 66.2%.
Interpretation:
Initially, there was a 50% chance of having picked the biased coin. After observing two consecutive heads (which is more likely with the biased coin than with the fair coin), our belief that we have the biased coin increases from 50% to approximately 66.2%. The evidence (two heads) supports the hypothesis of a biased coin more strongly than the hypothesis of a fair coin.
Consider the Priors: How would the result P(Biased | Two Heads) change if your prior belief was that the biased coin was much rarer (e.g., P(Biased) = 0.1 and P(Fair) = 0.9)?