The mean and variance of the Binomial distribution are given by the following formulas: "\\mu=np" and "\\sigma^2=np(1-p)" .
5 trials means that "n=5" .
Substitute this value into two formulas:
"\\mu=5p"
"\\sigma^2=5p(1-p)"
Since the sum of the mean and variance is "9\/5" :
"5p+5p(1-p)=9\/5"
This is a quadratic equation that we need to solve in order to find "p"
"5p+5p-5p^2=9\/5"
"-5p^2+10p=9\/5"
Divide both sides by "-5" :
"p^2-2p=-9\/25"
Complete the square:
"p^2-2p+1=-9\/25+1"
"(p-1)^2=16\/25"
"p-1=-4\/5" or "p-1=4\/5"
"p=1\/5=0.2" or "p=9\/5=1.8"
Since the probability can't be greater than 1, the only solution is "p=0.2" .
Now, that we know the value of "p" and "n" , we can find "P(x \\ge 1)" . Since it's the complement to "P(x=0)" ,
"P(x \\ge 1)=1-P(x=0)"General formula for "P(x)" for Binomial distribution is "P(x)=C(n,x)p^x(1-p)^{n-x}" , thus:
Answer: "P(x \\ge 1)=0.67232" .
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