Let us consider a coin tossing experiment with an unfair coin, for which the probability of head is p (>0.5) at each trial. (1) Find the probability of three heads in five trials. (2) Assume that p = 0.55. Derive the probability that the number of heads is between 500 and 600 out of 1000 experiments. (3) Derive a lower bound of the probability obtained in (2) using Bernoulli's Theorem. (4) Do you think the probability in (2) can be approximated by Gaussian distribution? Justify your answer. (5) If your answer for the question (4) is yes, then calculate the probability in (2) using the normal approximation and the error function table given below. erf(x) X erf(x) 0.60 0.65 0.70 0.75 X 0.05 0.01994 0.80 0.28814 1.55 0.43943 2.30 0.10 0.03983 0.85 0.30234 1.60 0.44520 2.35 0.15 0.05962 0.90 0.31594 1.65 0.45053 2.40 0.20 0.07926 0.95 0.32894 1.70 0.45543 2.45 0.25 0.09871 1.00 0.34134 1.75 0.45994 2.50 0.30 0.11791 1.05 0.13683 1.10 0.35314 1.80 0.36433 1.85 0.35 0.40 0.15542 1.15 0.37493 1.90 0.45 0.17364 1.20 0.38493 0.50 0.19146 1.25 0.39435 erf(x) X 0.46407 2.55 0.46784 2.60 0.47128 2.65 1.95 0.47441 2.70 2.00 0.47726 2.75 0.55 0.20884 1.30 0.22575 1.35 0.24215 1.40 0.40320 2.05 0.41149 2.10 0.41924 2.15 0.25804 1.45 0.42647 2.20 0.27337 1.50 0.43319 0.47982 2.80 0.48214 2.85 0.48422 2.90 2.95 0.48610 2.25 0.48778 3.00 erf(x) 0.48928 0.49061 0.49180 0.49286 0.49379 0.49461 0.49534 0.49597 0.49653 0.49702 0.49744 0.49781 0.49813 0.49841 0.49865