How to randomly pick half* of the item of a list with equity**?

How to pick exactly^* half of the item of a list with equity^**?

^(*)For a list of n item, if n is even, half is n/2. if n is odd, half is n/2 + 1.

The number of item in the list is more than 10^9. So n/2+1 is not close enought to n to be a noticable approximation.

^(**)The principe of equity and fairness mean that every items of the list have the same probality of beeing pick.

I had a Equity Picker like this:

 myList.Where(i => rand.NextDouble() >= 0.5);

but with this I can have more/less than half, with a slight chance of all /none of them.

Disclaimer: this is not an home work more a reflexion and modelisation around Thanos Random Picker