Answer to Question #324408 in Statistics and Probability for Gaby

Question

Advanced QM

1. In the past year, Gino’s Pizza Restaurant was profitable on exactly 75% of its business days. During the next 8 business days, what is the probability that Gino’s will be profitable for:

(a) exactly 6 days?

(b) less than 7 days?

2. An insurance company says that 15% of all fires are caused by arson. A random sample of 50 fire insurance claims are under study. Let r be the number of fires that were really started by arson in this sample.

(a) What is the expected number (mean) of arson fires among these 50 fires?

(b) What is the standard deviation of the number of arson fires?

Answer 1

$1:\\X-number\,\,of\,\,profitable\,\,days\\X\sim Bin\left( 8,0.75 \right) \\a:P\left( X=6 \right) =C_{8}^{6}\cdot 0.75^6\cdot \left( 1-0.75 \right) ^{8-6}=0.311462\\b:P\left( X<7 \right) =1-P\left( X\geqslant 7 \right) =\\=1-C_{8}^{7}\cdot 0.75^7\cdot \left( 1-0.75 \right) ^{8-7}-0.75^8=0.632919\\2:\\X\sim Bin\left( 50,0.15 \right) \\a:EX=np=50\cdot 0.15=7.5\\b:\sigma =\sqrt{np\left( 1-p \right)}=\sqrt{50\cdot 0.15\cdot \left( 1-0.15 \right)}=2.52488$