TITLE: Should I throw the dice again if I have rolled 4?
QUESTION [8 upvotes]: My math skills are very basic so it might be a stupid question, I had a discussion with my brother in law and now we have a 'math problem'.
We were playing a game with dices and he threw 4. The challenge was to throw the highest number, you can stop or throw again once, you cant see what the opponent has thrown, you both reveal after finishing. He said if you have 4 you have a 50% change the next time to throw the same or higher. 
(4,5,6) vs (1,2,3) 
and should throw again. 
But I said that I won't throw again on 4 because you already threw 4 and you are not likely to throw 4, 2 times in a row and therefore I would stop at 4. Am I right or is he and do you have a 50 % chance on throwing 4 or higher again? 
Game rules

You throw a normal dice 1/6
You can choose to throw again or keep the current value
Your opponent cant see your value, you cant see his.
The one with the highest value wins.

Short version:
If I threw 4, how big is the chance I throw 4 or more the next time I throw the dice, and should i take that chance?.
Thoughts
Average of 1 dice is 3.5, if i throw 4 im above Average and am more likely to throw below average next time.

REPLY [2 votes]: He said if you have 4 you have a 50% change the next time to throw the same or higher.

This is true, but is misleading. It's also true that you have a 4/6 chance to throw the same or lower.
If you wish to maximize the expected value, this is the strategy to follow:
The expected value each time you roll is $3.5$ (since this is the average of the values, and they have equal probabilities). Since your roll of $4$ is above this, rerolling is expected to lower your score, not raise it. You should reroll $(1,2,3)$ and hold $(4,5,6)$. Note that the rolls are completely independent events, so rolling a 4 the first time does not decrease the probability that you'll roll a 4 the second time.
Whether this actually results in you beating your opponent more than 50% of the time depends on his strategy, which likely depends on what he knows of your strategy.