# Conditional Probability & Cards: Why P(S2)=13/52?

My course says that if you draw two cards from a deck, define the events: S1 = ‘first card is a spade’ and S2 = ‘second card is a spade’ then P(S2)=13/52. Why 13/52? My thinking is this:

P(S2) is the probability that the second card is a spade given that we do not know whether the first card is a spade or not. If the first card is a spade then the chance that the second card is a spade is 12/51. If the first card is not a spade then the chance is 13/51. So we have a 13/52 chance that P(S2) is 12/51 and we have a 39/52 chance that P(S2) is 13/51. How can you say P(S2)=13/52?

• Isn’t $\frac{13}{52}\frac{12}{51}+\frac{39}{52}\frac{13}{51}=\frac{13}{52}$ anyways?
– user700480
Jun 2, 2022 at 19:37

\begin{align} P(S_2) &= P(S_2\mid S_1)P(S_1) + P(S_2 \mid S_1^c)P(S_1^c) \\ &= \frac {12} {51} \times \frac {13} {52} + \frac {13} {51} \times \frac {39} {52} \\ &= \frac {13} {52} \left(\frac {12} {51} + \frac {13} {51} \times 3 \right) \\ &= \frac {13} {52} \end{align}