Hypothesis testing: Fisher's exact test and Binomial test My first post in this community. My knowledge of statistics is limited, and so I am seeking advice on the following problem. (Sorry for the long post, and thank you in advance for your help).
I have a group of 15 children with taste disorders (a not so common condition); 8 are girls, and 7 are boys.
I do the following experiment. Each child in my group tastes a slice of a specially prepared cake, and then I ask the child "does the cake taste good?", and the child can answer either "yes" or "no". Each child tastes the cake separately, and answer my question before meeting any other child in the group.
I collect the following data:





cake tastes good
cake does not taste good





girls
7
1
8


boys
6
1
7



13
2
15





*

*Considering the population of girls with tastes disorders, I do a binomial test with number of success k = 7, number of trials n = 8, and probability of success p = 0.5, to test my null hypothesis H0 = "my cake tastes good for no more than 50% of the population of girls with taste disorders". In python I can run binomtest(7, 8, 0.5, alternative="greater") which gives the following result BinomTestResult(k=7, n=8, alternative='greater', proportion_estimate=0.875, pvalue=0.03515625) and ConfidenceInterval(low=0.5293205913988617, high=1.0). I find that p-value <= 0.05, and therefore I can reject H0, and say that "my cake tastes good for more than 50% of the population of girls with taste disorders".


*Similarly, considering the population of boys with tastes disorders, I can do a binomial test to test my null hypothesis "my cake tastes good for no more than 50% of the population of boys with taste disorders". In python I can run binomtest(6, 7, 0.5, alternative="greater") which gives the following result BinomTestResult(k=6, n=7, alternative='greater', proportion_estimate=0.8571428571428571, pvalue=0.0625). I find that p-value > 0.05, and therefore I cannot reject H0, and I say that "my cake tastes good for no more than 50% of the population of boys with taste disorders".


*Now I run a Fisher's exact test on my contingency table. My null hypothesis is H0 = "there is no significant difference between the proportion of girls with taste disorder who find that my cake tastes good, and the proportion of boys with taste disorders who find that my cake tastes good". In python I can run fisher_exact([[7, 1], [6, 1]], alternative="two-sided") which gives the following result (1.1666666666666667, 1.0), where the fist value (1.17) is the odds ratio, and the second value (1) is the p-value. I find that p-value >= 0.05, and therefore I cannot reject the null hypothesis, and I say that "there is no significant difference between the proportion of girls with taste disorder who find that my cake tastes good, and the proportion of boys with taste disorders who find that my cake tastes good".
The result obtained with the Fisher's exact test ("no significant difference between the proportion of girls and boys who finds that the cake tastes good") seems to contradict the results in (1) and (2), which say that the "more than 50% of the population of girls find that the cake tastes good" (1), and "no more than 50% of boys find that the cake tastes good" (2). How do interpret these results?
[problem re-phrased (hopefully in a better way) according to whuber suggestion]

Narrowing the question on my own, I would
be interested whether, among children with taste
disorder, liking the taste of your cake is
independent of gender. [For example, if the disorder were
color-blindness, it might make sense to look
for gender differences.]
In R, Fisher's Exact test looks like this:
TBL = rbind(c(7,1), c(6,1))
fisher.test(TBL)

        Fisher's Exact Test for Count Data

data:  TBL
p-value = 1
alternative hypothesis: 
 true odds ratio is not equal to 1
95 percent confidence interval:
   0.01276832 104.39935860
sample estimates:
odds ratio 
  1.154699 

Here, Fisher's Exact test would be a traditional alternative
to a chi-squared test of independence on a
a $2 \times 2$ table. The table has too few
subjects disliking the cake to give a reliable
P-value using a chi-squared test:
chisq.test(TBL)

        Pearson's Chi-squared test 
        with Yates' continuity correction

data:  TBL
X-squared = 9.9048e-32, df = 1, p-value = 1

Warning message:
In chisq.test(TBL) : 
 Chi-squared approximation may be incorrect

Specifically, the warning message appears when
any of the expected counts in the chi-squared
test is below 5. Here are expected counts corresponding to your observed counts.
chisq.test(TBL)$exp
         [,1]      [,2]
[1,] 6.933333 1.0666667
[2,] 6.066667 0.9333333
Warning message:
In chisq.test(TBL) : 
 Chi-squared approximation may be incorrect

As implemented in R, one can sometimes get a more
useful P-value for the chi-squared test by simulation; simulation makes no difference
in the reported P-value for your data.
chisq.test(TBL, sim=T)

        Pearson's Chi-squared test 
        with simulated p-value 
        (based on 2000 replicates)

data:  TBL
X-squared = 0.010302, df = NA, p-value = 1

