# Binomial theorem vs probability of a sequence

In Daniel Kahneman's book "Thinking, Fast and Slow", he offers the following example of how our minds identify causal patterns out of random chance.

A hospital has 6 babies born in a day. We know that male and female births are equally likely and births are independent. Are these 3 sequences equally likely?

BBBGGG GGGGGG BGBBGB

Intuitively we don't think so, however, since the events are independent then all 3 sequences have equal probability (he doesn't state but I presume this is $$0.5^6$$).

My question is, how does this reconcile with the fact that the probability of having 6 girls out of 6 births from the binomial thereom is far less likely than 3 girls (0.015625 vs 0.3125)? If the question was asking about different sequences of the same ratio of G/B births then they would all have equal chance, but I can't see how it is true in its current form.

My question is, how does this reconcile with the fact that the probability of having 6 girls out of 6 births from the binomial thereom is far less likely than 3 girls (0.015625 vs 0.3125)?

The probability of 3 girls is the probability of bbbggg, bbgbgg, bbggbg, bbgggb, bgbbgg, bgbgbg, bgbggb,gbbbgg, gbbgbg, gbbggb, bgggbb, bggbgb, bggbbg, gbggbb, gbgbgb, gbgbbg, ggbgbb, ggbbgb, ggbbbg, and gggbbb. Each of these 20 has individually 0.015625 probability, and to get any of these you have 0.3125 probability.

Are these 3 sequences equally likely?

BBBGGG GGGGGG BGBBGB

Intuitively we don't think so, however...

I can see two reasons why we make these assessments of GGGGGG and BGBBGB different:

### Microstates and macrostates

There are a lot more ways how you can get 3 girls than just the BGBBGB sequence.

To get this particular sequence BGBBGB sequence is as likely as to get the sequence GGGGGGG. But when you are comparing the total number of girls and boys, should you care about the specific order?

Often one is not interested in the particular details but instead one is interested in the global numbers. For instance one is interested in the probability of a particular total number of boys and girls, and not in a specific order.

These distinctions between microstates and macrostates occur very often in physics. For instance, if you get a tire punctured then it will very likely loose pressure and get empty. However, microscopically a particular state of deflation might be just as likely as a particular state of remaining full.

### Inverse probability

Indirectly, when we are 'analyzing' a particular binary sequence, then we are not just looking at the probability of that sequence, and instead we look at the relative probability. This is because often we are not directly interested in the probability of the observation (not in the first place and only indirectly) but instead we are interested in what the observation tells about the underlying unknown parameters. And observations with equal probability (given a particular hypothesis) may still tell a different story (because they do not necessarily have equal probability for other hypotheses). It is not about the difference in probability for different observations, but it is about the difference in likelihood (or the likelihood-ratio) for different parameter values.

If the probability parameters for boys and girls are equal $$p_{boy}=p_{girl}=0.5$$ then the observation GGGGGG is not less or more unlikely than the observation BGBBGB. However the observation GGGGGG still stands out as a special observation because it is relatively more likely when $$p_{girl}$$ is higher. The observation GGGGGG is more likely when $$p_{girl}>p_{boy}$$ than when $$p_{girl}=p_{boy}$$, and therefore we single it out as a special case.

So in the end we are more often comparing/assessing those sequences by means of likelihood ratios.

An example of likelihood for the case of an ordered sample of 5 is below ('theta' is here the probability for a 'b'). You can see that only the total number is relevant (and not the order) for the likelihood (this relates to sufficiency).

observation        probability of observing given theta

bbbbb              (1-theta)^0(theta)^5

rbbbb              (1-theta)^1(theta)^4
brbbb              (1-theta)^1(theta)^4
bbrbb              (1-theta)^1(theta)^4
bbbrb              (1-theta)^1(theta)^4
bbbbr              (1-theta)^1(theta)^4

rrbbb              (1-theta)^2(theta)^3
rbrbb              (1-theta)^2(theta)^3
rbbrb              (1-theta)^2(theta)^3
rbbbr              (1-theta)^2(theta)^3
brrbb              (1-theta)^2(theta)^3
brbrb              (1-theta)^2(theta)^3
brbbr              (1-theta)^2(theta)^3
bbrrb              (1-theta)^2(theta)^3
bbrbr              (1-theta)^2(theta)^3
bbbrr              (1-theta)^2(theta)^3

rrrbb              (1-theta)^3(theta)^2
rrbrb              (1-theta)^3(theta)^2
rbrrb              (1-theta)^3(theta)^2
brrrb              (1-theta)^3(theta)^2
rrbbr              (1-theta)^3(theta)^2
rbrbr              (1-theta)^3(theta)^2
brrbr              (1-theta)^3(theta)^2
rbbrr              (1-theta)^3(theta)^2
brbrr              (1-theta)^3(theta)^2
bbrrr              (1-theta)^3(theta)^2

brrrr              (1-theta)^4(theta)^1
rbrrr              (1-theta)^4(theta)^1
rrbrr              (1-theta)^4(theta)^1
rrrbr              (1-theta)^4(theta)^1
rrrrb              (1-theta)^4(theta)^1

rrrrr              (1-theta)^5(theta)^0