Probability of getting x heads n times when tossing 100 times

Question

I have 3 coins, and I flip them all together and count the number of heads I get. I repeat this 100 times and write down how many times I get 0 heads, ..., 3 heads.

I want to prove that based on my results the coins are biased. For that I want to use the Chi-Square Test, but I need to write my null hypothesis and I don't know how.

From what I understand, the null hypothesis would be that the coins are unbiased. And then I could write something like this:

Every time I toss the coins I have $2^3$ possible outcomes
So the probability of getting $x$ heads on a toss is $P'(x) = \begin{pmatrix} 3 \\ x \end{pmatrix}/2^3$
Then if I do this 100 times, the probability of getting x heads, n times could be $P(n,x) = \begin{pmatrix} 100 \\ n \end{pmatrix}P'(x)^nP'(x)^{100-n}$

But I'm not sure if $P(n,x)$ is correct, is it?

If the formula was correct then I could use it to calculate the expected value and perform the Chi-Test.

$H_0$ claims that the coins are unbiased, hence the probability for zero or three heads is $1/8$ and for one or two heads $3/8$. — Michael Hoppe, Nov 01 '19 at 11:36
@MichaelHoppe thanks for reasuring that that should be the claim. I tried to get to a formula based on that but I'm not sure if it makes sense — user8804, Nov 02 '19 at 11:53

score 1 · Answer 1 · answered Nov 02 '19 at 12:22

Too long for a comment ...

If the coin was unbiased, we expect for the number of heads $$\begin{matrix} \text{number of heads}&0&1&2&3\\ \hline \text{expected numbers}& 12.5 & 37.5 & 37.5 & 12.5 \end{matrix} $$ Now, e.g., if the actual outcome was $$\begin{matrix} \text{number of heads}&0&1&2&3\\ \hline \text{observed numbers}& 10 & 30 & 40 & 20 \end{matrix} $$ calculate the test statistic $${\chi^2}^\ast=\biggl(\sum\frac{(\text{observed number})^2}{\text{expected number}}\biggr)- n$$ and compare it with the critical value $\chi^2_{1-\alpha,n-1}$.

Probability of getting x heads n times when tossing 100 times

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