2)Three cards are drawn at random from a standard deck without replacement. What is the probability that all three cards are hearts?
3)Three cards are drawn at random from a standard deck with replacement. What is the probability that exactly two of the three cards are red?
4)Three cards are drawn at random from a standard deck without replacement. What is the probability that exactly two of the three cards are red?
5)Three cards are drawn at random from a standard deck with replacement. What is the probability that at least two of the three cards are red?
I am having trouble finding the correct answer when I am having to do the math to figure out the with replacement and without replacement. And I also don't understand the difference in the math I need to do to figure out the exactly two of the three cars and the at least two cards.
Best Answer
The term without replacement means that the deck is different for each draw so the probability of getting a heart on the first draw is $\frac{13}{52}$ as there are 13 hearts in a full deck of 52 cards.
Now given you have already drawn a heart on your first draw the probability of getting a heart on your second draw is $\frac{12}{51}$ as there are now only 12 hearts left in the pack and the pack has 51 cards remaining.
So the probability of hearts on both your first and second draw is $\frac{13}{52} \cdot \frac{12}{51} = \frac{156}{2652} = \frac{1}{17}$
I'm sure you can see from here how you would calculate three hearts.
The term with replacement means that the card is put back and mixed up again after each draw so the probability of drawing a heart on the second draw is $\frac{13}{52}$ because you are still drawing from a full pack of cards.
For two hearts the probability is $\frac{13}{52} \cdot \frac{13}{52} = \frac{169}{2704} = \frac{1}{16}$ and for three hearts ...