Question

Two customers are visiting a particular shop in the same week (Monday to Saturday). Each day is equally likely to visit the shop on any one day as on another. What is the probability that both will visit the shop on:
(i) The same day
(ii) Different days
(iii) Consecutive days

Answer 1

Hint: We assume two variables denoting each customer and find the total possibilities of visiting a particular shop. We write the possibilities in the form of ordered pairs where we write the first element as a possibility for one customer and second element as a possibility for the second customer. Using the formula for probability which is the number of favorable outcomes divided by the total number of outcomes we find probability in each of three cases.

Complete step-by-step solution:
We have two customers. Let us denote one by A and second by B. Both customers visit the shop between the days Monday, Tuesday, Wednesday, Thursday, Friday and Saturday.
Total number of days of the week in which Customers visit the shop is 6.
Then we draw a table for possibilities of them visiting the particular shop on days of the week from Tuesday to Saturday.
We denote the possibilities as a set of ordered pairs \[(x,y)\] where x denoted the day A visited the shop and y denotes the day B visited the shop.
Here we denote the days by:
Monday-M, Tuesday -T, Wednesday -W, Thursday -TH, Friday -F and Saturday -S

(A) (B)	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
Monday	(M,M)	(T,M)	(W,M)	(TH,M)	(F,M)	(S,M)
Tuesday	(M,T)	(T,T)	(W,T)	(TH,T)	(F,T)	(S,T)
Wednesday	(M,W)	(T,W)	(W,W)	(TH,W)	(F,W)	(S,W)
Thursday	(M,TH)	(T,TH)	(W,TH)	(TH,TH)	(F,TH)	(S,TH)
Friday	(M,F)	(T,F)	(W,F)	(TH,F)	(F,F)	(S,F)
Saturday	(M,S)	(T,S)	(W,S)	(TH,S)	(F,S)	(S,S)

We see total possible outcomes are 36.
Now we solve each part separately.
(i) The same day:
In this case we take all possibilities where both A and B visit the shop on the same day.
So, we take a set of those ordered pairs from the table where both first and second elements are equal.
Favorable outcomes are: (M,M), (T,T), (W,W), (TH,TH), (F,F), (S,S)
So the number of favorable outcomes is 6.
We use the formula for probability of an event which is given by the number of favorable outcomes divided by total number of outcomes.
Probability that A and B visit shop on same day \[ = \dfrac{6}{{36}}\]
Cancel out common factors from numerator and denominator.
Probability that A and B visit shop on same day \[ = \dfrac{1}{6}\]
(ii) Different days:
Since we know the probability of A and B visiting the shop on the same day is \[\dfrac{1}{6}\].
So, the probability of A and B visiting the shop on different days will be \[1 - \]probability of A and B visiting the shop on the same day.
So, probability of A and B visiting the shop on different days \[ = 1 - \dfrac{1}{6}\]
Taking LCM we can write
Probability \[ = \dfrac{{6 - 1}}{6}\]
Solving the fraction we get
Probability of A and B visiting the shop on different days \[ = \dfrac{5}{6}\].
(iii) Consecutive days:
In this case we take all possibilities where both A and B visit the shop on consecutive days.
So, we take a set of those ordered pairs from the table where both first and second elements are consecutive to each other. Consecutive means side by side or one day ahead or behind.
Favorable outcomes are: (M,T),(T,W), (W,TH), (TH,F), (F,S), (T,M), (W,T),(TH,W), (F,TH), (S,F)
So number of favorable outcomes is 10
We use the formula for probability of an event which is given by the number of favorable outcomes divided by the total number of outcomes.
Probability that A and B visit shop on consecutive days \[ = \dfrac{{10}}{{36}}\]
Cancel out common factors from numerator and denominator.
Probability that A and B visit shop on consecutive days \[ = \dfrac{5}{{18}}\]

Note: Students can many times make the mistake of leaving out some of the total outcomes when they try to write the outcomes roughly, they are advised to make use of the table to write the outcomes which later helps in selection of favorable outcomes as well.
Also, keep in mind the answer of probability should always be greater than or equal to zero and less than or equal to one.