Question

A salesman has the 70% chance to sell a product to any customer. The behaviour of the successive customers is independent. If two customers A and B enter, what is the probability that the salesman will sell the product to customer A or B?
(a) 0.98
(b) 0.91
(c) 0.70
(d) 0.49

Answer 1

Hint: First, before proceeding for this, we must write the information clearly given in the question that both the customers are independent and their probability is given by
$\begin{align}
  & P\left( A \right)=\dfrac{70}{100} \\
 &\Rightarrow P\left( A \right)=0.7 \\
\end{align}$ and $\begin{align}
  & P\left( B \right)=\dfrac{70}{100} \\
 & \Rightarrow P\left( B \right)=0.7 \\
\end{align}$.
Then, we are given with the condition that both the customers are independent which gives the condition as $P\left( A\cap B \right)=P\left( A \right)\times P\left( B \right)$. Then, by using the formula from the sets for the probability of A or B is given by $P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$, we get the desired answer.

Complete step-by-step answer:
In this question, we are supposed to find the probability that the salesman will sell the product to customer A or B when the salesman has the 70% chance to sell a product to any customer and the behaviour of the successive customers is independent.
So, before proceeding for this, we must write the information clearly given in the question that both the customers are independent and their probability is given by:
$\begin{align}
  & P\left( A \right)=\dfrac{70}{100} \\
 & \Rightarrow P\left( A \right)=0.7 \\
\end{align}$ and $\begin{align}
  & P\left( B \right)=\dfrac{70}{100} \\
 & \Rightarrow P\left( B \right)=0.7 \\
\end{align}$
Now, we are given with the condition that both the customers are independent which gives the condition as:
$P\left( A\cap B \right)=P\left( A \right)\times P\left( B \right)$
So, by substituting the value of the P(A) and P(B) in the above expression, we get:
$\begin{align}
  & P\left( A\cap B \right)=0.7\times 0.7 \\
 & \Rightarrow P\left( A\cap B \right)=0.49 \\
\end{align}$
Then, by using the formula from the sets for the probability of A or B is given by:
$P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$
Then, by substituting all the values in the above expression, we get:
$P\left( A\cup B \right)=0.7+0.7-0.49$
Then, by solving the above expression, we get the probability of A or B as:
$\begin{align}
  & P\left( A\cup B \right)=1.4-0.49 \\
 & \Rightarrow P\left( A\cup B \right)=0.91 \\
\end{align}$
So, we get the desired probability as 0.91

So, the correct answer is “Option (b)”.

Note: Now, to solve these types of the questions we need to know some of the basics of union and intersection also known as OR and AND operators. So, the union is the collection of sets is the set of all elements in the collection and intersection of two sets A and B is the set containing all the elements which are common in both A and B.