How long would it take to see every possible video poker hand?

I am asking this just out of random curiosity...

After studying and playing strategies for a few different variants of video poker, I started wondering if anyone has actually seen every possible combination of cards available for "decision making" in video poker. By this, I mean viewing all 2,598,960 possible 5-card poker hands (not all possible permutations). If they were presented one by one, it would take about 4332 hours to see them all at 600 hands per hour, which would certainly be doable in a couple of years worth of "full time" play. However, the cards are presented randomly, so one would see certain combinations multiple times and others less often. I'm not sure to calculate this -- how long would one have to play in order to have (for example) 95% confidence that they have seen every card? My hunch is that it would be a very long time.

About 26 million hands. The form for the answer is sum for i from 1 to 2598960 of (2598960/i)

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Quote: epaulson

I'm not sure to calculate this -- how long would one have to play in order to have (for example) 95% confidence that they have seen every card? My hunch is that it would be a very long time.

Nice question.

This looks to be a coupon collector problem with 2,598,960 coupons.

Recently over at 2+2 forum a few posters, including BruceZ and Sherman,
went crazy with this problem and solution, in a good way,
and wrote some R code to calculate the Mean, Quantiles , Variance and Standard Deviation for up to a large number of coupons.

If interested in reading
http://forumserver.twoplustwo.com/25/probability/expected-number-trials-all-195-numbers-chosen-range-1-195-a-1278005/index5.html

I tested it (R code) out and compared the values to my Excel.
(yes I did a few years ago when I first got Excel 2007)
The results from the R program matched up to 1 million coupons,
so I continued.

That made me think the Wizard's answer could be a bit low, maybe,
but his solution is correct for the average number of hands to play.

Results (The R program is fast)
Mean = 39,888,416 (rounded)

My Excel: 39,888,416.48

The R program results
> couponv2(2,598,960)
Mean, Median, VAR, SD
Results
39,888,416, 39,340,806, 1.111082e+13, 3,333,289

Now for your 95% confidence interval: 46,107,666 hands
> couponv2(2,598,960, .95)
Mean, Quantile 0.95, VAR, SD
Results
39,888,416, 46,107,666, 1.111082e+13, 3,333,289

at 99%: 50,343,850 hands
> couponv2(2,598,960, .99)
Mean, Quantile 0.99, VAR, SD
Results
39,888,416, 50,343,850, 1.111082e+13, 3,333,289


I agree to your "hunch is that it would be a very long time"

Quote: 7craps

My Excel: 39,888,416.48

Got this too for the mean number of hands. I'm sure BruceZ's work is on the money; the man is simply amazing.

Thanks for the answers everyone! So it seems, that after 20-30 years of playing, a few die-hard veteran players of video poker could reasonably expect to have seen every possible hand.