Answer:
a) Â [tex]P(10)=0.2061[/tex]
b) Â [tex]P(9\ or\ 12)=0.3172[/tex]
c) Â [tex]P(X>10)=0.51545[/tex]
Step-by-step explanation:
From the question we are told that:
Probability of Same day resolution [tex]P=70\%=>0.7[/tex]
Probability of Another day resolution [tex]Q=1-P=>30%=>0.3[/tex]
Sample size [tex]n=15[/tex]
a)
Generally the equation for Probability 10 of the problems can be resolved today is mathematically given by
Since
 [tex]P(X)=^{n}C_{x}*p^{x}*Q^{n-x}[/tex]
 [tex]P(10)=^{15}C_{10}*p^{10}*Q^{15-10}[/tex]
 [tex]P(10)=\frac{15!}{10!(15-10)!*0.70^{10}*0.30^5}[/tex]
 [tex]P(10)=0.2061[/tex]
b)
Generally the equation for Probability 9 or 12 of the problems can be resolved today is mathematically given by
 [tex]P(9)=\frac{15!}{9!(15-9)!*0.70^{9}*0.30^6}[/tex]
 [tex]P(9)=5005*2.942*10^{-5}[/tex]
 [tex]P(9)=0.1472[/tex]
OR
 [tex]P(12)=\frac{15!}{12!(15-12)!*0.70^{12}*0.30^3}[/tex]
 [tex]P(12)=455*3.737*10^{-4}[/tex]
 [tex]P(12)=0.17[/tex]
Therefore Probability 9 or 12 of the problems can be resolved today is
 [tex]P(9\ or\ 12)=P(9)+P(12)[/tex]
 [tex]P(9\ or\ 12)=0.1472+0.17[/tex]
 [tex]P(9\ or\ 12)=0.3172[/tex]
c)Generally the equation for probability more than 10 of the problems can be resolved today is mathematically given by
 [tex]P(X>10)=P(11)+P(12)+P(13)+P(15)[/tex]
Where
 [tex]P(11)=\frac{15!}{11!(15-11)!*0.70^{11}*0.30^4}[/tex]
 [tex]P(11)=0.2186[/tex]
 [tex]P(13)=\frac{15!}{13!(15-13)!*0.70^{13}*0.30^2}[/tex]
 [tex]P(13)=0.0916[/tex]
 [tex]P(14)=\frac{15!}{14!(15-14)!*0.70^{14}*0.30^1}[/tex]
 [tex]P(14)=0.0305[/tex]
 [tex]P(15)=\frac{15!}{15!(15-15)!*0.70^{15}*0.30^0}[/tex]
 [tex]P(15)=0.00475[/tex]
Therefore
 [tex]P(X>10)=0.2186+0.0916+0.0305+0.00475+0.17[/tex]
 [tex]P(X>10)=0.51545[/tex]
