Probability of drawing the ace of spaces exactly one time in 17 trials?

Given a regular pack of 52 playing cards, I know that...

1)... the probability of randomly drawing the Ace of Spades each time, over 17 trials, is
(1/52) ^ 17 (6.72889E-30)

2)... the probability of NOT drawing the Ace of Spaces all 17 times is...
(51/52) ^ 17 (about .7188)

3)... the probability of drawing the Ace of Spaces at LEAST once, in 17 attempts, is...
1 - (51/52) ^ 17 (about .2811)

But what is the probability of drawing the Ace of Spaces EXACTLY one time, in 17 trials?

I'm having a brain fart and can't seem to figure that out. A simulation tells me the answer should be real close to .2397, but I can't come up with the formula to arrive at that figure.

Thanks for the help.

51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 51/52 * 1/52 * 17 = x

x = 0.23961549411564

That's it.

Ah, closer to .2396.

Thanks.

You're welcome.

Quote: EdCollins

But what is the probability of drawing the Ace of Spaces EXACTLY one time, in 17 trials?

You asked for the binomial distribution probability
There are many online calculators

http://en.wikipedia.org/wiki/Binomial_distribution
shows the formula

p=1/52
n=17
k=1
Combin(N,K)= n!/k!(n-k)!

Combin(17,1) * p^1 * (1-p)^17-1

P= 17*(1/52)*(51/52)^16

short and long versions look good
Enjoy

If these examples are for education, one should explain the formulas a bit.

Drawing the Ace of Spades in (all of) one attempts is (1/52)^1.
Not drawing the Ace of Spades in 16 attempts is (51/52)^16.

These numbers you figured out, great work. Now to come to your originial question.
There are 17 ways to draw the Ace of Spades exactly once in 17 attempts.
On the first trails, on the second trail, ... on the last trail.

Say, we need the probability drawing the Ace of Spades only the first trail of 17 trails.
This must be (1/52)^1 * (51/52)^16 from the above numbers, because you are drawing
the Ace of Spades in all of one trail, and then not drawing the Ace in all of 16 trails.

Now lets look at the probability drawing the Ace of Spades only on the second of 17 trails.
You guess it, it's again (1/52)^1 * (51/52)^16.

Since there are 17 distinct ways to draw the Ace of Spades exactly once in 17 trails,
your answer is 17 * (1/52)^1 * (51/52)^16.


The next question you should try is: what is the probability of drawing the Ace of Spades exactly twice in 17 trails.
You can check your result with the formula of guido, but you should try to do without it.