The probability of getting a head when a coin is tossed 1 time: p=0.5.
Let's find the mean:
μ=np=400⋅0.5=200.
The standard deviation:
σ=np(1−p)=400⋅0.5⋅(1−0.5)=100=10.
(a) The probability of obtaining between 185 and 210 heads inclusive:
P(185≤X≤210)=P(184.5<X<210.5).
Let's find z-scores of 184.5 and 210.5:
z1=10184.5−200=10−15.5=−1.55,
z2=10210.5−200=1010.5=1.05.
Now we have to use z-table.
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).
Let's find z-scores of 204.5 and 205.5:
z1=10204.5−200=104.5=0.45,
z2=10205.5−200=105.5=0.55.
Now we have to use z-table.
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<175.5)+P(x>227.5).
Let's find z-scores of 175.5 and 227.5:
z1=10175.5−200=10−24.5=−2.45,
z2=10227.5−200=1027.5=2.75.
Then we have:
P(X<175.5)+P(x>227.5)=P(z<−2.45)+P(z>2.75)=
=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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