diff --git a/_problems/unit-01/I-law-of-total-probability-and-Bayes-rule/2.md b/_problems/unit-01/I-law-of-total-probability-and-Bayes-rule/2.md index e6803a5..a523e59 100644 --- a/_problems/unit-01/I-law-of-total-probability-and-Bayes-rule/2.md +++ b/_problems/unit-01/I-law-of-total-probability-and-Bayes-rule/2.md @@ -38,19 +38,19 @@ First, we find the probability of getting heads $(P(H))$. Since one coin always Since each coin is equally likely to be chosen, each has a $1/3$ probability of being chosen. Thus, $P(H)$ is: \begin{align} -\[ P(H) = P(C_{hh}) \cdot P(H|C_{hh}) + P(C_{tt}) \cdot P(H|C_{tt}) + P(C_{f}) \cdot P(H|C_{f}) \] -\[ P(H) = \frac{1}{3} \cdot 1 + \frac{1}{3} \cdot 0 + \frac{1}{3} \cdot 0.5 \] -\[ P(H) = \frac{1}{3} + 0 + \frac{1}{6} \] -\[ P(H) = \frac{1}{2} \] +\[ P(H) &= P(C_{hh}) \cdot P(H|C_{hh}) + P(C_{tt}) \cdot P(H|C_{tt}) + P(C_{f}) \cdot P(H|C_{f}) \]\\\\ +&= \frac{1}{3} \cdot 1 + \frac{1}{3} \cdot 0 + \frac{1}{3} \cdot 0.5 \]\\\\ +&= \frac{1}{3} + 0 + \frac{1}{6} \]\\\\ +&= \frac{1}{2} \] \end{align} Now, using Bayes' Theorem, P(A|H) is: - -\[ P(C_{hh}|H) = \frac{P(H|C_{hh}) \cdot P(C_{hh})}{P(H)} \] -\[ P(C_{hh}|H) = \frac{1 \cdot \frac{1}{3}}{\frac{1}{2}} \] -\[ P(C_{hh}|H) = \frac{1}{3} \times 2 \] -\[ P(C_{hh}|H) = \frac{2}{3} \] - +\begin{align} +\[ P(C_{hh}|H) &= \frac{P(H|C_{hh}) \cdot P(C_{hh})}{P(H)} \]\\\\ +& = \frac{1 \cdot \frac{1}{3}}{\frac{1}{2}} \]\\\\ +& = \frac{1}{3} \times 2 \]\\\\ +& = \frac{2}{3} \] +\end{align} So, the probability that the coin is the two-headed coin, given that it landed on heads, is 2/3. **(b) Probability that the same coin will land on heads in the next flip**