# Law of total probability

Suppose you had a normal deck of 52 playing
cards and lost a card. You then decide to draw
a card from the remaining 51 cards.

What is the probability the drawn card is a spade?

Would this be appropriately captured by the following
events:
A : event card was drawn from the deck
S : event card drawn is a spade
S^c : event card drawn is not a spade

then

P(A) = P(A | S)P(S) + P(A|S^c)P(S^c)

But this is annoying me because
if we called S "event card drawn is a spade"
and A "event card was drawn from deck"
then P(A | S) doesn't seem to make much
sense to me. That is, "event card drawn
from the deck given drawn card is a spade"
is pretty much incoherent.

What am I missing or how can I resolve this issue?

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tiny-tim
Homework Helper
hi hholzer!
Suppose you had a normal deck of 52 playing
cards and lost a card. You then decide to draw
a card from the remaining 51 cards.

What is the probability the drawn card is a spade?

Would this be appropriately captured by the following
events:
A : event card was drawn from the deck …
(btw, that's not the way we use the word "event" )

I don't understand what your A is supposed to be

You want P(S) …

split it up into P(S|lost card was a spade) and P(S|lost card was not a spade)

(are you sure you've copied the question correctly? it seems obvious the answer is 0.25 )

Ah, that's what I was trying to determine. So we break it up into

The answer is indeed 1/4 but I was more concerned
with how we partition the sample space.

And on the word "event", "event" is a subset of your sample space,
as you of course know. The three events would be:

S = {card randomly drawn from deck of 51 cards is a spade }
A = {lost card is a spade }
A^c = { lost card is not a spade }