Question

1-1

A venture capitalist invests in startup technology companies. She expects about 12.5% of these companies to be successful. Complete parts (a) through (c) below (a) What is the probability that she will have to invest in more than seven companies before she comes upon one that is successful? Be sure to state any assumptions that you make The probability is (Round to four decimal places as needed.)

Answer 1

Here, probability of company is successful is 0.125 i.e p=0.125 and it is same for each company.

Let the random variable X is

X: Number of companies in which a venture invests getting a successful one.

X~Geometric(p=0.125)

Assumption:

• Each observation falls into one of two categories: Success or Failure (or whatever you wish to call them).
• The probability of success is the same for each observation.
• The observations are independent.

Now , the probability that she will have to invest in more than 7 companies before she comes upon one that is successful is

$\\P(X>7)=1-P(X\le7) \\ \\=1-(Pr(X=1)+Pr(X=2)+Pr(X=3)+Pr(X=4)+Pr(X=5)+Pr(X=6)+Pr(X=7)) \\ \\= 1-\{(1-0.125)^{ 1 - 1} \times 0.125 + (1-0.125)^{ 2 - 1} \times 0.125 + (1-~~~0.125)^{ 3 - 1} \times 0.125 + (1-0.125)^{ 4 - 1} \times 0.125 + (1-~~0.125)^{ 5 - 1} \times 0.125 + (1-0.125)^{ 6 - 1} \times 0.125 + (1-~~0.125)^{ 7 - 1} \times 0.125\} \\ \\= 1-\{0.125 + 0.1094 + 0.0957 + 0.0837 + 0.0733 + 0.0641 + 0.0561\} \\ \\=1-0.6073 \\ \\=0.3927$
$\\P(X>7)=1-P(X\le7) \\ \\=1-(Pr(X=0)+Pr(X=1)+Pr(X=2)+Pr(X=3)+Pr(X=4)+Pr(X=5)+Pr(X=6)+Pr(X=7)) \\ \\= 1-\{(1-0.125)^{ 0} \times 0.125 + (1-0.125)^{ 1} \times 0.125 + (1-0.125)^{ 2} \times 0.125 + (1-0.125)^{ 3} \times 0.125 + (1-0.125)^{ 4} \times 0.125 + (1-0.125)^{ 5} \times 0.125 + (1-0.125)^{ 6} \times 0.125+(1-0.125)^{ 7} \times 0.125\} \\ \\= 1-\{0.125 + 0.1094 + 0.0957 + 0.0837 + 0.0733 + 0.0641 + 0.0561+0.0491\} \\ \\=1-0.6564 \\ \\=0.3436$

=0.6073