SOLUTION: I am trying to get a simplified breakdown of how to solve this equation without using a calculator? 56% of men consider themselves professional baseball fans. You randomly selec

Question 1073050: I am trying to get a simplified breakdown of how to solve this equation without using a calculator?
56% of men consider themselves professional baseball fans. You randomly select 10 men and ask each if he considers himself a professional baseball fan. Find the probability that the number who consider themselves baseball fans is (a) exactly five, (b) at least six, and (c) less than four.

Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
Without a calculator, this is difficult, because even the approximation of 50% is not too accurate.
One could use a normal approximation which barely makes the assumptions valid.
E(x)=5.6, and V(x)=5.6*0.44=0.2464, almost 0.25, so sd is 0.5
5 would be about 1 sd to the left (1.2 sd exactly) and about 0.31 or a little more
at least 6 would be everything to the right of z=0.8, which is a little less than 0.75.
less than 4 would be z< 3.1 sd and this would be expected to be near 0. The true values are considerably different:
The exact number is 10C5(0.56)^5(0.44)^5
This is not too dissimilar from getting 5 heads in 10 coin tosses.
That is 252*(1/1024) or about a quarter. The exact answer is 0.2289
For at least 6, it would more, but the values of 5,4,3,2,1,0 would not be relevant 0.2427+0.1765+0.0843+0.0238+0.003, or 0.5276
For less than 4, it would be 3,2,1,0 and those probabilities are roughly 175/1024, a little less, since the probability of 0.44 is less than 0.50 that I am using to estimate.
0.0002+0.0035+0.0198+0.0672+=0.0907
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