Video Transcript
If three cards are drawn from an
ordinary deck of 52 playing cards without replacement and the first two cards are
not red, find the probability that the third card is a diamond.
We’re told that three cards are
drawn from the deck without replacement, and the first two are not red. Given this information, we’re asked
to find the probability that the third card is a diamond. We’re therefore interested in how
many diamonds are left in the deck by the time we get to the third card.
At this point, we should recall
that a standard deck of 52 playing cards contains four suits: hearts and diamonds,
which are red, and clubs and spades, which are black. There are 13 cards in each
suit. So, if the first two cards drawn
are not red, this also means they are not diamonds.
We begin then with the full deck of
52 cards, 13 of which are diamonds. The first card is then drawn, and
we’re told it’s not a diamond. As the card is not replaced in the
deck, there are now 51 cards remaining. As the card chosen was not a
diamond, all 13 diamonds remain in the deck. Then, the second card is drawn,
which we’re told is also not a diamond. So there are now 50 cards
remaining, but all 13 diamonds still remain.
Finally, the third card is
drawn. As there are now 50 cards in the
deck and 13 of them are diamonds, the probability that we now choose a diamond is 13
out of 50. We can express this conditional
probability as the probability the third card is a diamond, given that the first two
cards are not red.
So, by considering how many
diamonds are left in the deck after the first two cards have been drawn, we’ve found
that the probability the third card is a diamond, given that the first two cards are
not red, is 13 over 50.
