Question

A shopkeeper has five customers who cycle on rent. He has three cycles and the probability that a customer will hire a cycle is $\dfrac{3}{4}$​. If he charges rs.2 for a cycle as a rent, the probability that he earns exactly Rs.6 per day is $\dfrac{{45K}}{{512}}$​. Find the value of k.

Answer 1

Hint : To find the value of k we need to find the probability that shopkeeper can earn exactly Rs 6 per day then equate it with the given probability in terms of k. to find the probability that shopkeeper can earn exactly Rs 6 per day first calculate the probability of not getting hired as probability of getting hired is already given.

Complete step-by-step answer :
Suppose A is the event that the cycle is hired
So the probability that the cycle is hired, $P\left( A \right) = \dfrac{3}{4}$​,
So the probability of cycle not getting hired can be calculated by subtracting P(A) in 1
$P(\overline A ) = 1 - P(A) \\\ P(\overline A ) = 1 - \dfrac{3}{4} \\\ P(\overline A ) = \dfrac{1}{4} \\\$
To earn Rs.6 per day his all the three cycles must be hired.
His cycles can be hired by 5 customers.
Hence the required probability ${ = ^5}{C_3} \cdot {\left( {\dfrac{{3}}{4}} \right)^3} \cdot {\left( {\dfrac{1}{4}} \right)^2} = \dfrac{{135}}{{512}}$
Now to find the value of k equate the above result with the given value of probability
We get,
$\dfrac{{45k}}{{512}} = \dfrac{{135}}{{512}} \\\ k = \dfrac{{135}}{{45}} \\\ k = 3 \\\$
Hence the value of k is equal to 3

Note : Many random experiments that we carry have only two outcomes that are either failure or success. For example, a product can be defective or non-defective, etc. These types of independent trials which have only two possible outcomes are known as Bernoulli trials. For the trials to be categorized as Bernoulli trials it must satisfy these conditions:
A number of trials should be finite.
The trials must be independent.
Each trial should have exactly two outcomes: success or failure.
The probability of success or failure remains unchanged for each trial.
Here in this question it is only asked for exactly Rs 6 per day earning so we will not use the concept of at least or at most