Suppose that a fair coin is tossed n times Estimate the prob

Suppose that a fair coin is tossed n times. Estimate the probability that the proportion of heads obtained lies between 0.49 and 0.51 for n - 100, 200, 500, 1000, and 2000.

Solution

By using Binomial Distribution Table , X = No. of Heads Obtained

(a) n = 100 , No of Heads lies between 49 and 51

=> P( 49 < X < 51 ) = P( X > 49 ) - P( X 51 )

=> P( 49 < X < 51 ) = 0.5398 - 0.4602 = 0.0796

(b) n = 200 , No of Heads lies between 98 and 102

=> P( 98 < X < 102 ) = P( X > 98 ) - P( X 102 )

=> P( 98 < X < 102 ) = 0.58396 - 0.41603 = 0.16793

(c) n = 500 , No of Heads lies between 245 and 255

=> P( 245 < X < 255 ) = P( X > 245 ) - P( X 255 )

=> P( 245 < X < 255 ) = 0.65632 - 0.34368 = 0.31264

(d) n = 1000 , No of Heads lies between 490 and 510

=> P( 490 < X < 510 ) = P( X > 490 ) - P( X 510 )

=> P( 490 < X < 510 ) = 0.72601 - 0.27399 = 0.45202

(e) n = 2000 , No of Heads lies between 980 and 1020

=> P( 980 < X < 1020 ) = P( X > 980 ) - P( X 1020 )

=> P( 980 < X < 1020 ) = 0.80841 - 0.19159 = 0.61682