Question

Seven chits are numbered 1 to 7. Three are drawn one by one with replacement. What is the probability that the least number on any selected chit is 5?
A. \[1 - {\left( {\dfrac{2}{7}} \right)^4}\]
B. \[4{\left( {\dfrac{2}{7}} \right)^4}\]
C. \[{\left( {\dfrac{3}{7}} \right)^3}\]
D. None of these

Answer 1

Hint First we will find the outcomes that are greater than or equal to 5. Then apply the formula probability to get one of the favorable outcomes. Since we will draw 3 chits one by one with replacement. Thus multiply the probability of getting favorable outcomes three times to get the required probability.

Formula used
\[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\]

Complete step by step solution:
Given that, seven chits are numbered 1 to 7.
The chits that are greater than or equal to 5 are chit 5, chit 6, chit 7.
Total number of chits is 7.
The number of chits that are at least 5 is 3.
Apply the formula \[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\] to calculate the probability of getting chit 5 or chit 6 or chit 7
The probability of getting chit 5 or chit 6 or chit 7 is \[ = \dfrac{3}{7}\].
Given that, three chits are drawn one by one with replacement.
Each drawing is independent. Since the probability of getting chit 5 or chit 6 or chit 7 is \[\dfrac{3}{7}\] and each draw we want to get a chit at least a number 5. So, the probability of each event is the same is \[\dfrac{3}{7}\].
The events are independent, so multiply \[\dfrac{3}{7}\] three times to get the required probability.
The required probability is \[\dfrac{3}{7} \times \dfrac{3}{7} \times \dfrac{3}{7} = {\left( {\dfrac{3}{7}} \right)^3}\].
Hence option C is the correct option.

Note: Many students make a common mistake to find favorable outcomes. They do not count chit 5 as favorable. We have to find the probability that the least number on any selected chit is 5. In this case, chit 5 is included in the favorable outcomes.