One way to estimate a is to do the following. Suppose you draw a square and its inscribed circle on a piece of paper. Then you randomly throw small grains one by one (or needles that can piece through...

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Extracted text: One way to estimate a is to do the following. Suppose you draw a square and its inscribed circle on a piece of paper. Then you randomly throw small grains one by one (or needles that can piece through the paper – the idea is that you throw very small items of which the shape can be ignored). After many throws, you count the proportion of grains/needles that end up inside the circle, multiply it by 4, and that's your estimate of T. An illustration is as follows: How would you use i.i.d. random variables to model the location the grain/needle throws? Write your estimate of T in terms of these i.i.d. random variables. Prove that under the i.i.d. model, your estimate of T converges in probability to T.