Answer to Question #347225 in Statistics and Probability for shiba

Question #347225

5. 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



1
Expert's answer
2022-06-06T08:34:04-0400

The probability of getting a head when a coin is tossed 1 time: p=0.5.p=0.5.

Let's find the mean:

μ=np=4000.5=200.\mu=np=400\cdot0.5=200.

The standard deviation:

σ=np(1p)=4000.5(10.5)=100=10.\sigma=\sqrt{np(1-p)}=\sqrt{400\cdot 0.5 \cdot (1-0.5)}=\sqrt{100}=10.


(a) The probability of obtaining between 185 and 210 heads inclusive:

P(185X210)=P(184.5<X<210.5).P(185\leq X \leq 210)=P(184.5 < X < 210.5).

Let's find z-scores of 184.5 and 210.5:

z1=184.520010=15.510=1.55,z_1=\frac{184.5-200}{10}=\frac{-15.5}{10}=-1.55,

z2=210.520010=10.510=1.05.z_2=\frac{210.5-200}{10}=\frac{10.5}{10}=1.05.

Now we have to use z-table.

P(184.5<X<210.5)=P(1.55<z<1.05)=0.85310.0606=0.7925.P(184.5 < X < 210.5)=P(-1.55<z<1.05)=0.8531-0.0606=0.7925.


(b) The probability of obtaining exactly 205 heads:

P(X=205)=P(204.5<x<205.5).P(X=205)=P(204.5<x<205.5).

Let's find z-scores of 204.5 and 205.5:

z1=204.520010=4.510=0.45,z_1=\frac{204.5-200}{10}=\frac{4.5}{10}=0.45,

z2=205.520010=5.510=0.55.z_2=\frac{205.5-200}{10}=\frac{5.5}{10}=0.55.

Now we have to use z-table.

P(204.5<x<205.5)=P(0.45<z<0.55)=0.70880.6736=0.0352.P(204.5<x<205.5)=P(0.45<z<0.55)=0.7088-0.6736=0.0352.


(c) The probability of obtaining fewer than 176 or more than 227 heads:

P(X<176orx>227)=P(X<175.5orx>227.5)=P(X<176 \: or \: x>227)=P(X<175.5 \: or \: x>227.5)=

=P(X<175.5)+P(x>227.5).=P(X<175.5)+P(x>227.5).

Let's find z-scores of 175.5 and 227.5:

z1=175.520010=24.510=2.45,z_1=\frac{175.5-200}{10}=\frac{-24.5}{10}=-2.45,

z2=227.520010=27.510=2.75.z_2=\frac{227.5-200}{10}=\frac{27.5}{10}=2.75.

Then we have:

P(X<175.5)+P(x>227.5)=P(z<2.45)+P(z>2.75)=P(X<175.5)+P(x>227.5)=P(z<-2.45)+P(z>2.75)=

=0.0071+(10.9970)=0.0071+0.003=0.0101.=0.0071+(1-0.9970)=0.0071+0.003=0.0101.


Answer: (a) 0.7925 (b) 0.0352 (c) 0.0101


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