Question

# Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. Find the probability that both will visit the shop on (a) the same day?(b) consecutive days?(c) different days?

Hint: While solving this problem of probability, we start with finding the total possible number of outcomes in this problem. The number of days from Tuesday to Saturday (including Tuesday and Saturday) is 5. Further, we calculate the outcome for two people (Shyam and Ekta), the total outcomes are $5\times 5=25$.

To make this clear, the possible outcomes are listed below
(T, T) (T, W) (T, Th) (T, F) (T, S)
(W, T) (W, W) (W, Th) (W, F) (W, S)
(Th, T) (Th, W) (Th, Th) (Th, F) (Th, S)
(F, T) (F, W) (F, Th) (F, F) (F, S)
(S, T) (S, W) (S, Th) (S, F) (S, S)
Where,
T stands for Tuesday
W stands for Wednesday
Th stands for Thursday
F stands for Friday
S stands for Saturday
We can clearly see that the listed outcomes are 25.
(a) the same days
From the list of outcomes, we can clearly see that for Ekta and Shyam to visit the particular shop on the same day, the desirable number of outcomes are 5. These are (T, T), (W, W), (Th, Th), (F, F), (S, S). Now,
\begin{align} & \text{probability}=\dfrac{\text{desired number of outcomes}}{\text{total number of outcomes}} \\ & \text{probability}=\text{ }\dfrac{5}{25} \\ & \text{probability}=\text{ }\dfrac{1}{5} \\ \end{align}
(b) consecutive days
From the list of outcomes, we can clearly see that for Ekta and Shyam to visit the particular shop on consecutive days, the desirable number of outcomes are 8. These are (T, W), (W, T), (W, Th), (Th, W), (Th, F), (F, Th), (F, S), (S, F).
\begin{align} & \text{probability}=\dfrac{\text{desired number of outcomes}}{\text{total number of outcomes}} \\ & \text{probability}=\text{ }\dfrac{8}{25} \\ \end{align}
(c) different days
To solve this, we simply need to subtract the probability for Ekta and Shyam to visit on the same days from 1. To explain this,
Probability (same days) + Probability (different days) = 1
Since, when Ekta and Shyam don’t visit a particular shop on the same day, they visit the particular shop on different days. Thus, the sum of their probabilities should add up to 1.
\begin{align} & \text{probability}=\text{ }1-\dfrac{1}{5} \\ & \text{probability}=\text{ }\dfrac{4}{5} \\ \end{align}

Note: A common mistake while calculating the probability for Ekta and Shyam to visit on consecutive days is to forget in counting all the desirable number of outcomes. To explain, one can forget to count (T, W), (W, T) as two separate outcomes and thus land up with the desired number of outcomes as 4. (namely, (T, W), (W, Th), (Th, F), (F, S)). This may lead to the calculation of incorrect probability.