If that’s the way you interpret it, then yes, we’d “expect” to see it. But your lead-in talked about runs in a series of flips and “almost guaranteed” so I assumed, you were implying it was very likely.
In any case, the way to figure out the probability of having a run of at least lenght r in n coin flips is given here:
Nitpick: meiotic drive can occur on any chromosome, and thus can occur in males as well as females, and does not refer exclusively to sex ratio biases. Further, a female-biased sex ratio is caused by the male carrying a lone driver X in some of his sperm, which then suppress/out-compete his Y-bearing sperm. Most sex ratio drive described thus far in Drosophila is X-chromosome based and thus female based. However, male-biased sex ratios have been described in mosquitoes and others. As a side note, for drive to be seen phenotypically (manifesting as sex ratio bias or other ways) the Y chromosome must carry a sensitive target, and there must also be a paucity of suppressing loci in relation to the driver/inducing loci.
It’s interesting that at-birth sex ratios differ by country per Wikipedia. I wonder if there is/are sex ratio drive elements in people that differ by population? Cool stuff. [/geek] 
I think we should all give BetsQ a big hand for providing this thread with Data (and plenty of it)
Further rampant speculation. According to that data set, the existence of older brothers very slightly increases the chance that the next kid will be a boy.
I Have Also Heard (on this very board, so lets assume it’s true) that the existence of older brothers very slightly increases a boy’s chance of being gay.
Connection?