The question: Suppose that three cards are drawn from a deck of cards with replacement, what is the probability of getting one or more hearts?

I think of the question as asking P(one heart or two hearts or three hearts), then I find P(one heart) and P(two heart) and finally P(three hearts), then add them up.

Here's what I did:

P(one heart) = (13/52)*(39/52)*(39/52)*3 (because the heart can appear as the second card or the third)
P(two heart) = (13/52)*(13/52)*(39/52)*3 (similar reason)
P(three heart) = (13/52)^3 * 3 (similar reason)

and found ~0.61 chance of getting one or more hearts.

Is that correct? I'm just not sure about what I did.

Thank you!

I made a probability tree.

The tree came out to have 64 different suite combination, and only 37 of those combination had the correct suite.

The odds then were

37/64 or ~ 57.8 percent.

You can also use the "binomial probability" form.

P(x)=C\large\begin{pmatrix}n\\x\end{pmatrix}p^xq^{ n-x}

Where C\large\begin{pmatrix}n\\x\end{pmatrix}=\frac{n!}{ (n-x)!x!}

Where x is the number of successes we want, n is the number trials, p is the chance of success, and q is the chance of failure.

For this question we have

P(1)+P(2)+P(3) (Because we want the odds of 1 heart, 2 hearts, and 3 hearts)

x = 1.. 2... 3, n=3, p=0.25, q=0.75

P(1)+P(2)+P(3) = \frac{3!}{(3-1)!1!}(0.25)^1(0.75)^{3-1}+\frac{3!}{(3-2)!2!}(0.25)^2(0.75)^{3-2}+\frac{3!}{(3-3)!3!}(0.25)^3(0.75)^{3-3}

simplified we have

P(1)+P(2)+P(3) = 3(0.25)(0.75)^{2}+3(0.25)^2(0.75)+(0.25)^3

And this also ends up being about 57.8 percent

There is only 1 way to have three hearts so you should not have multiplied by three in calculating P(three heart)!

Ok, my method:

P(no heart) = (39/52)^3

P(at least a heart) = 1 - (39/52)^3

Shorter, simpler! :)

P(three heart) = (13/52)^3 * 3 (similar reason)


My previous post was to point out the error in the above line of reasoning given in the original question, the * 3 part is incorrect as there is only one way to have three hearts. I did not mean that the question wanted a three of hearts. I used the same notation that kyva used.

Nhatkiem please consider re rating my post as it is not incorrect!

My previous post was to point out the error in the above line of reasoning given in the original question, the * 3 part is incorrect as there is only one way to have three hearts. I did not mean that the question wanted a three of hearts. I used the same notation that kyva used.

Nhatkiem please consider re rating my post as it is not incorrect!

I see, sorry, you didn't quote what you were referring to, I thought you were talking about P(3) alone. Unfortunately it won't allow me to change the rating at this time.

However I will try to track of your post in the future for rating when I am allowed to rate for you again.