@San:
You have 42.1% chances of getting at least one technology by the 3rd roll. (The probability of missing on your 3 first rolls is (5/6)3 = 0.579 or 57.9% so you will have at least one tech 42.1% of the time.)
By the way Gamerman, what was exactly your question?
My question was when should one expect to hit their first tech, on average, when rolling 1 die per round. In other words, at one point does one get to a 50/50 chance of getting a tech?
So as you pointed out, after 3 attempts, on average you would have between 1 and 3 techs 42.1% of the time. After 6 attempts, you would have between 1 and 6 techs about 2/3 of the time. (So I don’t know why you and Allweneed are saying you could expect to have a tech on your 6th roll, on average) After 4 rolls, you have over 50% chance of hitting between 1 and 4 techs. These stats are all assuming you buy one die at a time.
In fact, I hit 4 straight techs with the Axis in a game I’m playing from round 2 into round 3. So my point is, you have a 100% of getting zero tech if you never buy a research die and if you never buy a research die you are not projecting the threat of hitting an instant tech to your opponent, who can play less defensively in accordance with the lack of threat. :-)