Video Transcript
Two cards are drawn from an
ordinary deck of 52 playing cards without replacement. Find the probability that the
second card is a king given that the first card is a king.
As this question concerns two
events, which are the two cards being drawn from a deck, we may find it helpful to
use a tree diagram to illustrate the situation. We’re given some information about
whether or not the first card is a king and asked to find the probability that the
second card will be a king. So we’ll use the outcomes of king
and not king for the tree diagram.
The first card could either be a
king or not a king. And then we have the same two
outcomes for the second card. We then need to find the
probabilities for each branch of the tree diagram. And at this point, it will be
helpful to recall that an ordinary deck of cards contains 52 cards, four of which
are kings. The probability that the first card
is a king is therefore four over 52, and the probability that the first card is not
a king is 48 over 52. Notice that although these
probabilities can be simplified, we will keep them with denominators of 52 because
this will help us when calculating the probabilities for the second card.
Now, we are told in the question
that the two cards are selected without replacement. This means that the outcome of
choosing the second card is dependent on the outcome of choosing the first. We therefore need to consider what
the outcome of the first event was when calculating the probability for the
second. As one card has been removed from
the deck and not replaced, there are now 51 cards remaining. So we can write the probabilities
for the second card as fractions with denominators of 51.
If the first card was a king, then
there are now only three kings remaining in the deck. So the probability the second card
is a king in this instance is three over 51. All 48 nonkings remain in the
deck. So the probability that the second
card will not be a king if the first was a king is 48 over 51. If the first card was not a king,
however, then all four kings remain in the deck. So the probability that the second
card will be a king if the first wasn’t is four over 51. The number of cards that are not
kings has reduced by one. So the probability that the second
card will not be a king if the first was also not a king is 47 over 51.
Notice that the probabilities on
each set of branches of the tree diagram sum to one. Now, let’s use our tree diagram to
answer the question, which was to find the probability that the second card is a
king given that the first card is a king. This conditional probability can be
found on the second set of branches of the tree diagram. If the first card was a king, the
probability that the second card will also be a king is three over 51. We can express this probability
using the conditional probability notation of a vertical line.
So, by drawing a tree diagram, we
found that when two cards are drawn from an ordinary deck without replacement, the
probability that the second card is a king given that the first is a king is three
over 51.
