What is the probability that the first card is spades given that the second and third cards are spades?
Attempt: Use Bayes' rule to calculate $P(E_1|E_2E_3)$
- $E_1$ = first card is spade
- $E_2$ = second card is spade
- $E_3$ = third card is spade
$$ P(E_1) = \frac{13}{52} $$ $$ P(E_2) = \frac{13}{52} $$ $$ P(E_3) = \frac{13}{52} $$
$$ P(E_1|E_2E_3) = \frac {P(E_2E_3|E_1)P(E_1) }{ P(E_2E_3)} $$ So this $$= \frac{(\frac{12}{51} \cdot \frac{12}{51})(\frac{13}{52})} {\frac{13}{52} \cdot \frac{13}{52}} = 0.2214$$
Is this correct?