What is the chance a given gene will end up in a given gamete? - (Aug/12/2018 )

Let us say a somatic cell had a desired gene. This somatic cell was replicated during interphase so that it had two of the desired gene. It then underwent meiosis.

My question then is, what is the chance a given gene will end up in a given gamete that will be involved in conception?

Firstly, there is a 50% chance a chromosome will be assigned to a given daughter cell. This is because of independent assortment. In addition, cross over occurs at chiasmata. Lets assume then that there was a 50% chance that a gene will be crossed over. 

The process during meiosis is that cross-over occurs first. We have two desired genes, each on one of two pairs of chromatid attached to each other. There is 50% chance for a given gene to cross over. For example, the chance for both desired genes to cross over is 1/4 (1/2 x 1/2), the chance only one will cross over is 2/4 (since either of the two desired genes can cross over).

Then there is the first independent assortment during meiosis I which produces the desired daughter cell. In this case there is a 50% chance that a given chromatid pair (containing the desired gene) will be assigned to the desired daughter cell. However, in the case where a desired gene is found in both homologs (2/4 of a chance), then the chance that at least one desired gene will be in the daughter cell is 100%.

Finally, during meiosis II, another independent assortment occurs which produces the desired gamete. The same rule applies with a 50% chance a given gene will be assigned to a given gamete. However, if the desired daughter cell contains two of the desired genes, then there is a 100% chance that the gamete will have the desired gene.

In conclusion, the chance I estimated that the gamete will have at least one desired gene is 50%. I'm not sure if this is correct, however. Anyone mind confirming this.

50% is correct. even with crossover, you still have a 50% chance of selecting a chromosome with desired version of the gene (you don't increase the number of chromosomes available, which would change the probability).