[Math] Probability of Multiple Weighted Coin Flips

poisson distributionprobabilitystatistics

I would like to calculate the probabilities of the outcomes of three weighted coins being flipped. I believe what I am looking for is a Poisson Binomial Distribution.

I am having trouble verifying/interpreting the results that I am finding on an online calculator.

Edit: Order does not matter in this question – the below is a table of the sums of outcomes.

+---------+---------+-------------+
| Outcome | Heads P | Probability |
+---------+---------+-------------+
| 3 heads |     .75 | .421875??   |
| 2 heads |     .75 | ??          |
| 1 heads |     .75 | ??          |
| 0 heads |     .75 | ??          |
|         |         |  (Sum = 1)  |
+---------+---------+-------------+

The .42 is calcualted for X>=3, but since there are only 3 flips it cannot be any greater. An alternate calculator provides a much lower answer, .03, which seems too low.

  1. Is a Poisson binomial distribution the correct calculation for this answer? (X=3 trials, .75 success)
  2. How would I find the probability of 2 out of 3 heads, 1 out of 3 heads, and no heads?

Thank you for taking the time to explain what I might be missing here.

Best Answer

If the coins all have the same bias (0.75 for showing heads), then the distribution for the count of heads among three flips is Binomial.

$$X\sim\mathcal{Binomial}(3,0.75)\quad\iff\quad\mathsf P(X{=}x) ~=~ \binom{3}{x}\,0.75^x\,0.25^{3-x}\,\mathbf 1_{x\in\{0,1,2,3\}}$$

(Poisson Binomial is when the coins each have a distinct bias).

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