On average, how many times does one need to roll a fair die so that the sum of the numbers rolled is at least 3?
At most 3 throws are needed for the sum of the numbers to be at least 3, so it is sufficient to look at the results of the first 2 throws. The first throw can result in 6 different outcomes, and the second throw, independent of the first, can also result in 6 different outcomes. This totals $6 \cdot 6=36$ possibilities.

Now let's examine how many cases there are where exactly 2 throws are needed to achieve a total of at least 3.

If a 1 appears on the first die, then at least a 2 must appear on the second die, so the successful throws are: $(1,2) ;(1,3)$; $(1,4) ;(1,5) ;(1,6):$ this totals 5 possibilities.

If a 2 appears on the first die, then whatever is thrown on the second die, the sum will be at least 3. This is 6 possibilities.

A single throw is sufficient if the score is at least 3, which happens in 4 cases. The second throw can then be anything.

This totals $4 \cdot 6=24$ possibilities.

Finally, 3 throws are needed in only one case, when the values of the first two throws are both 1.

Thus, the total number of throws is: $1 \cdot 3+11 \cdot 2+24 \cdot 1=49$. The average number of throws (or the average throw length) over 36 throws is $\frac{49}{36} \approx 1.36$.