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kyva1929 · Oct 19, 2009, 08:04 PM

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!

Nhatkiem · Oct 19, 2009, 10:14 PM

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.

Nhatkiem · Oct 19, 2009, 11:07 PM

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

elscarta · Oct 20, 2009, 03:25 AM

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

Unknown008 · Oct 20, 2009, 04:27 AM

Ok, my method:

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

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

Shorter, simpler! :)

elscarta · Oct 21, 2009, 12:25 AM

Originally Posted by kyva1929

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!

Nhatkiem · Oct 21, 2009, 12:29 AM

Originally Posted by elscarta

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.