Janelle draws one card
from a standard deck of 52 cards.
Determine the probability of drawing either
a queen or a king, then determine the probability
of drawing either a 10 or a club.
In this video, we will determine
the probabilities using the appropriate formula
below based upon whether the events
are mutually exclusive or not mutually exclusive.
So in the first example we're asked to find
the probability of drawing either a queen or a king.
Let's indicate this as the probability of Q
for queen or K for king.
Because it's not possible for a card to be both
a queen and a king at the same time, the events
of selecting a queen and selecting a king
are called mutually exclusive where again
two events are mutually exclusive or disjoint
if they cannot occur at the same time.
When this is the case, we can find
the probability of A or B by determining
the sum of the probability of A
and the probability of B.
So again because a card cannot be both a queen
and a king at the same time, the probability
of a queen or a king is equal to the probability
of a queen plus the probability of a king.
So because we have 52 total cards, four of which
are queens, the probability of selecting a queen
is equal to four out of 52 or 4/52.
Plus the probability of a king, and since
there are four kings there are four favorable
outcomes out of a total of 52 outcomes
the probability of selecting a king is also 4/52.
Adding, we get 8/52, which is a fraction
that can be simplified.
The greatest common factor between eight and 52 is four.
We divide both by four to simplify, which gives
us a simplified probability of 2/13.
And let's also show the decimal and percent.
2/13 to four decimal places is approximately 0.1538
which is equal to 15.38%.
And just to make sure this makes sense
when determining the probability of selecting
a queen or a king, either a queen or a king
is a favorable outcome and there are eight cards
in the deck that are queens or kings,
four queens and four kings, so we do have a total
of eight favorable outcomes out of a total
of 52 outcomes, which is shown here
in the probability, which does simplify to 2/13.
And now let's determine the probability
of drawing either a 10 or a club.
Let's indicate this by the probability of a 10 or a club.
In this case because a card can be both a club
and a 10 at the same time, the events of drawing
a 10 and drawing a club are not mutually exclusive.
So going back to our notes just for a moment
two events are not mutually exclusive
if they can both occur at the same time.
When events are not mutually exclusive
the probability of A or B is equal
to the probability of A plus the probability
of B minus the probability of A and B.
So going back to our example because a card
can be both a 10 and a club, the probability
of a 10 or a club is equal to the probability
of a 10 plus the probability of a club
minus the probability of a 10 and a club.
And let's determine these probabilities.
For the probability of selecting a 10,
since there are four 10s in the deck
there are four favorable outcomes out
of a total of 52 cards.
The probability of selecting a 10 is 4/52
plus the probability of a club.
Well, there are 13 clubs in the deck
and therefore there are 13 favorable outcomes
out of a total of 52 outcomes.
So we have plus 13/52.
And then minus the probability of a 10 and a club.
There's only one card that's a 10 and a club
which is the 10 of clubs.
So we have one favorable outcome out
of a total of 52 outcomes.
So we have minus 1/52.
Let's just stop here for a second and see why
we have to subtract this probability here.
When determining the probability of a 10
we had four favorable outcomes, these outcomes here.
And then when determining the probability
of a club, we had these 13 favorable outcomes.
But notice how by adding these favorable outcomes
we counted the 10 of clubs twice
and we only should count it once
which is why we have to subtract the probability
of a 10 and a club.
And again, that's because the two events
are not mutually exclusive because it is possible
for a card to be both a 10 and a club.
The common denominator here is 52
and the numerator is four plus 13 minus one, which is 16.
And going back to the cards one more time
we have a total of 13, 14, 15, 16 favorable
outcomes to have a card that is a 10 or a club
which is why this probability is correct.
To simplify we divide both by four
which gives us 4/13.
And let's go ahead and find the decimal and percent.
To four decimal places, we have approximately 0.3077
which is 30.77%.
I hope you found this helpful.