What is the probability that a randomly chosen card from a deck of 52 cards will be a queen or spade?

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You deal 5 cards from a well-shuffled deck of playing cards. What is the probability that the 5th card is the queen of spades?

Just from analysis, P(5th queen spade) = (51*50*49*48*1)/(52*51*50*49*48) = 1/52

However why wont this method of logic thinking incorrect?

P(5th queen spade) = (51Cr4) / (52Cr5) = 5/52. Reasoning: choose any first 4 cards and last card is queen spade, divide by all possible choice

Whenever you solve a probability question involving two conditions, and you are being asked to find the probability that either will occur for a given action, you are looking for what is known as a "union probability". Formally speaking, if we say #A# represents "the card is a Spade", and #B# represents "the card is a Queen", then we are looking for the probability of "the card is a Spade or a Queen", or symbolically:

#P(A uu B)#

The trick here is that these two possible events are not disjoint events; in other words, it can be possible to pull a single card and have it be a Spade and a Queen at the same time. The formula for determining #P(A uu B)# takes this into consideration:

#P(A uu B) = P(A) + P(B) - P(A nn B)#

(This is read as "the probability of A union B is equal to the probability of A plus the probability of B minus the probability of the intersection of A and B".)

If we consider #P(A)# (the probability the card is a Spade), in a standard deck of 52 cards there are exactly 13 cards which are Spades. Thus, #P(A) = 13/52 = 1/4#. (This is intuitive, because there are 4 suits of cards with the same values/ranks within them and we're only interested in one of those four suits.)

If we consider #P(B)# (the probability the card is a Queen), in a standard deck of 52 cards there are exactly 4 cards which are Queens (in suits of Hearts, Spades, Clubs, and Diamonds). Thus, #P(B) = 4/52 = 1/13#. (Again, this is intuitive, because there are 13 unique values of cards, of which there is only one Queen value.)

However, the probability #P(A nn B)# represents the probability the card is a Spade and a Queen at the same time. Of all 52 cards in the deck, there is only one Queen of Spades, thus #P(A nn B) = 1/52#.

Thus:

#P(A uu B) = P(A) + P(B) - P(A nn B)#

# = 13/52 + 4/52 - 1/52 = 16/52 = 4/13#