The expected value of the sum S of the weights of the 100 coins in the sample is now E(S) = n* E(S) = 12770 gr, with SE(S)=10 gr. As before the long run histogram of S is normal, centered at 12770 gr and with variance equal to 100 (10 squared).
P(survive) = P(12768 < S < 12832) = P( (12768-12770)/10 < Z < (12832-12770)/10 ) = P( -.2 < Z < 6.2 )which is approximately 58 %. Therefore, his chances of surviving are better than 1 in 2.
However, if he manages to survive he will keep a lot of gold. It will take him n * E(X) = 12,770,000 gr to manufacture the 100,000 coins, give or take SD * (square root of n) = 1 * (square root of 1000,000) = 316 gr. He will receive 100000 * 128 = 12,800,000 gr to manufacture them, so he will end up with 12,800,000 - 12,770,000 = 30,000 gr, give or take about 316 gr.
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