I Calculating the odds of drawing 3 "no" notes from a hat with 3 "no" and 3 "yes" notes

[Forwardbite1]

TL;DR Summary

My wife and I would like to know what the odds are of a "hat draw" we did this past week.

Hello All,

I would like to find out what the percentage is of what happened to us in a drawing - quite amazing and would like to know what the percentage is in doing this.

My wife and I were looking into starting another business but it looked like we were getting roadblocked at every turn. My wife had an idea and thought we should draw out of a hat to let us know if we were to go ahead with this new business by her and I drawing the same answer in each of our (3) draws - (6) draws total.

She put (3) pieces of paper in a hat that had "yes" on them, and she put (3) pieces of paper in the hat that had "no" on them.

My wife drew (3) times and got "no" on all (3) draws.

**Keep in mind that after each of her (3) draws, the piece of paper she drew was NOT put back into the hat. It was set aside on the counter. In other words, prior to her first draw, there were (3) "yes" pieces of paper and (3) "no" pieces of paper in the hat. After her first draw there were (3) "yes" pieces of paper and only (2) "no" pieces of paper in the hat. After her second draw there were (3) "yes" pieces of paper and only (1) "no" piece of paper in the hat. When I drew, I used the same method of setting the piece of paper that I drew to the side before doing my next draw.

My wife drew (3) "no" pieces of paper. The (3) "no" pieces of paper were then put back in the hat and I then I also drew (3) "no" pieces of paper, while also using the same method of setting aside the drawn piece of paper on the counter and not back in the hat. Each draw having varying odds since the drawn piece of paper was not put back in the hat. Also, not sure if this would be somehow calculated, but we did also draw (6) "no" in a row.

Could someone please calculate what the percentage is of this happening? I hope I've provided enough info for someone (someone much better at math than me! Ha!), but feel free to let me know if any additional info is needed for this calc..

Thank You!!!

 

[PeroK]

Not a whit, we defy augury. Hamlet

The probability of drawing all three "no" is $\frac 3 6 \times \frac 2 5 \times \frac 1 4 = \frac 1 {20}$. The probability of both doing it is, therefore, $\frac 1 {400}$.

 

[Office_Shredder]

One important thing to note for statistical purposes, you would be here making the same thread and asking the same question if you both got all yesses so I think PeroK's answer, while correct, underestimates the important thing which is the probability that you are amazed by what happened and think it has cosmic meaning.

 

[Forwardbite1]

We got the answer we were looking for (1/400), and thank you for providing this answer.