Law of total probability

[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
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?

[tiny-tim]

hi hholzer! :smile:
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" :wink:)

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

You want P(S) …

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

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

[hholzer]

Ah, that's what I was trying to determine. So we break it up into
(Lost card was spade) and (Lose card not spade).

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 }