Answer:
Step-by-step explanation:
P
(
Spade
∪
Queen
)
=
4
13
Step-by-step explanation:
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
∪
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
∪
B
)
takes this into consideration:
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
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
∩
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
∩
B
)
=
1
52
.
Thus:
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
=
13
52
+
4
52
−
1
52
=
16
52
=
4
13
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