Question 714753: According to a recent study, 1 in every 6 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled 25 women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that more than 22 of the women sampled have not been the victim of domestic abuse.
Would like to see the formula that gives the answer 0.188687
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let X = number of women who have not been the victim of domestic abuse
So each trial has a probability of 1 - 1/6 = 5/6 = 0.83333333333333
So
p = 0.83333333333333
n = 25
Now compute the following probabilities P(X = 23), P(X = 24), P(X = 25)
P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
P(X = 23) = (25 C 23)*(0.83333333333333)^(23)*(1-0.83333333333333)^(25-23)
P(X = 23) = (25 C 23)*(0.83333333333333)^(23)*(0.16666666666667)^(25-23)
P(X = 23) = (300)*(0.83333333333333)^(23)*(0.16666666666667)^2
P(X = 23) = (300)*(0.015094938254969)*(0.0277777777777789)
P(X = 23) = 0.125791152124747
P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
P(X = 24) = (25 C 24)*(0.83333333333333)^(24)*(1-0.83333333333333)^(25-24)
P(X = 24) = (25 C 24)*(0.83333333333333)^(24)*(0.16666666666667)^(25-24)
P(X = 24) = (25)*(0.83333333333333)^(24)*(0.16666666666667)^1
P(X = 24) = (25)*(0.0125791152124741)*(0.16666666666667)
P(X = 24) = 0.0524129800519766
P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
P(X = 25) = (25 C 25)*(0.83333333333333)^(25)*(1-0.83333333333333)^(25-25)
P(X = 25) = (25 C 25)*(0.83333333333333)^(25)*(0.16666666666667)^(25-25)
P(X = 25) = (1)*(0.83333333333333)^(25)*(0.16666666666667)^0
P(X = 25) = (1)*(0.0104825960103951)*(1)
P(X = 25) = 0.0104825960103951
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To summarize, we found the following
P(X = 23)= 0.125791152124747
P(X = 24)= 0.0524129800519766
P(X = 25)= 0.0104825960103951
Now add up the individual probabilities to find P(X > 22)
P(X > 22) = P(X = 23)+P(X = 24)+P(X = 25)
P(X > 22) = 0.125791152124747+0.0524129800519766+0.0104825960103951
P(X > 22) = 0.18868672818711
So the probability that more than 22 of the women sampled have not been the victim of domestic abuse is roughly 0.18868672818711
If you want to round to 6 decimal places, you would get 0.188687
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