Dependant Probability - Theoretical and Experimental have different outcomes?

#1
Ok, I hope someone might be able to offer me some assistance into why i'm seeing a difference between what i believe to be the correct method to calculate dependant probability.

Scenario: Coin bag of 8 weighted coins containing coins weighted for heads (0.5, 0.5, 0.6, 0.6, 0.7, 0.9, 0.5, 0.5) so thats 8 coins. I want to know the probability of picking HHT, picking one out at a time without replacement.

coin.png

I've calculated a theoretical probability of 12.7%

cal.png

BUT I've got some VBA knowledge so in excel I wrote some code to do the experiment and every time it comes out around 14.6% even when i've left it running for 500,000 tries it still never works out to be what I expect.

Is there something wrong with my code or is my calculation wrong?

Any help would be appreciated.

Lew
 
#2
I think your calculation is incorrect. It doesn't seem to exhaust all possible 3-coin selection paths, for example 0.6 --> 0.5 --> 0.9. A probability tree may prove helpful. Assuming the coins are otherwise indistinguishable apart from their bias, there are 43 distinct ways to select a 3-coin sequence without replacement under the conditions stated by the problem, and each such way has its own associated probability of being picked.

The analytical answer to the problem is P(HHT) = 1021/7000 ≈ 0.145857.
 
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