A coin is tossed 400 times. use the normal curve approximation to find the probability of obtaining (a) between 185 and 210 heads inclusive; (b) exactly 205 heads; (c) fewer than 176 or more than 227 heads.

Solution:

It is given that

A coin is tossed 400 times

Considering coin to be fair we have p = 0.5

E(x) = np = 400 (0.5) = 200

Variance (X) = npq = 200 (0.5) = 100

Standard deviation (x) = √Variance = 10

So the binomial approximated to normal after checking the conditions X is N (200, 100)

As the discrete is changed to continuous distribution, continuity correction should be done

So the probability of obtaining

(a) between 185 and 210 heads inclusive

P (184.5 < x < 210.5) = P (-1.55 < z < 1.05)

= 0.4394 + 0.3531

= 0.7925

(b) exactly 205 heads

P (204.5 < x < 210.5) = P (0.45 < z < 0.55)

= 0.2088 - 0.1736

= 0.0352

(c) fewer than 176 or more than 227 heads

P (X < 176.5 + P (X > 226.5)) = P (Z < -2.35) + P (Z > 2.65)

= 0.0094 + 0.0040

= 0.0134

Therefore, the probability of obtaining between 185 and 210 heads inclusive is 0.7925, exactly 205 heads is 0.0352 and fewer than 176 or more than 227 heads is 0.0134

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A coin is tossed 400 times. use the normal curve approximation to find the probability of obtaining (a) between 185 and 210 heads inclusive; (b) exactly 205 heads; (c) fewer than 176 or more than 227 heads.

Summary:

A coin is tossed 400 times. Using the normal curve approximation, the probability of obtaining

(a) between 185 and 210 heads inclusive is 0.7925

(b) exactly 205 heads is 0.0352

(c) fewer than 176 or more than 227 heads is 0.0134