I'll just repeat what I said before. Play the game a few dozen times and it will be clear that the odds of winning double (or within experimental error of that) if you switch. Then go through your own logic and think about every step in turn looking for the (false) assumption where you decide the odds must be equal. If we try to point one out for you, we will simply get into an argument about whether or not that step is valid, or can be rephrased, or otherwise shoe-horned to fit so a similar assumption resurfaces somewhere else: much better if you spot your own error, because you will be alert for similar assumptions you make on other occasions.