Question

The probability that a new small business closes before the end of its first year is 42%. In addition, 37% of all new businesses are started by women. The probability that a new business is either owned by a woman or goes out of business is 62%. Your sister starts a new business. What is the probability her business will still open at the end of the first year? Answer: A = A new small business closes before the end of its first year. B = New business started by a woman.

Answer 1

Answer:

The probability that my sister's business will still open at the end of the first year is 0.541.

Step-by-step explanation:

The events are denoted as:

A = A new small business closes before the end of its first year.

B = New business started by a woman.

It is provided that:

P (A) = 0.42

P (B) = 0.37

P (A ∪ B) = 0.62

The probability that my sister's business will still open at the end of the first year is expressed as $P(\bar A|B)=\frac{P(\bar A\cap B)}{P(B)}$.

The value of $P(\bar A\cap B)=P(B)-P(A\cap B)$.

Compute the value of P (A ∩ B) as follows:

$P(A\cap B)=P(A) + P(B) - P(A\cup B)=0.42+0.37-0.62=0.17$

The value of $P(\bar A\cap B)$ is:

$P(\bar A\cap B)=P(B)-P(A\cap B)=0.37-0.17=0.20$

Compute the value of $P(\bar A|B)$ as follows:

$P(\bar A|B)=\frac{P(\bar A\cap B)}{P(B)}=\frac{0.20}{0.37}=0.54054\approx0.541$

Thus, the probability that my sister's business will still open at the end of the first year is 0.541.