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.

What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?

Answer 1

Probability:

The probability of an event is the fraction of the number of times that the event can occur to the possible number of outcomes.

$\mathrm{Probability}\mathrm{of}\mathrm{an}\mathrm{event}=\frac{\mathrm{Number}\mathrm{of}\mathrm{outcomes}\mathrm{for}\mathrm{that}\mathrm{event}}{\mathrm{Number}\mathrm{of}\mathrm{possible}\mathrm{outcomes}}$

Total number of days from Tuesday to Saturday$=5$

Number of ways that Shyam can visit the shop$=5$

Number of ways that Ekta can visit the shop$=5$

Possible number of outcomes, $\mathrm{n}\left(\mathrm{S}\right)=5\times 5=25$

(i) Calculating the probability that both will visit the shop on the same day:

Consider $\mathrm{A}$ being an event that both will visit the shop on the same day.

$\mathrm{P}\left(\mathrm{A}\right)$ be the probability that both will visit the shop on the same day$\mathrm{P}\left(\mathrm{A}\right)=(\mathrm{Tuesday},\mathrm{Tuesday}),(\mathrm{Wednesday},\mathrm{Wednesday}),(\mathrm{Thursday},\mathrm{Thursday}),(\mathrm{Friday},\mathrm{Friday}),(\mathrm{Saturday},\mathrm{Saturday})$

$\mathrm{n}\left(\mathrm{A}\right)=5$

$\mathrm{P}\left(\mathrm{A}\right)=\frac{\mathrm{n}\left(\mathrm{A}\right)}{\mathrm{n}\left(\mathrm{S}\right)}=\frac{5}{25}=\frac{1}{5}$

Hence, the probability that both will visit the shop on the same day is $\frac{\mathbf{1}}{\mathbf{5}}$.

(ii) Calculating the probability that both will visit the shop on consecutive days:

Consider $\mathrm{B}$ being an event that both will visit the shop on consecutive days.

$\mathrm{P}\left(\mathrm{B}\right)$ be the probability that both will visit the shop on consecutive days.

$$\begin{gathered} \mathrm{P}\left(\mathrm{B}\right)=(\mathrm{Tuesday},\mathrm{Wednesday}),(\mathrm{Wednesday},\mathrm{Tuesday}),(\mathrm{Wednesday},\mathrm{Thursday}),(\mathrm{Thursday},\mathrm{Wednesday}),(\mathrm{Thursday},\mathrm{Friday}), \\ (\mathrm{Friday},\mathrm{Thursday}),(\mathrm{Friday},\mathrm{Saturday}),(\mathrm{Saturday},\mathrm{Friday}) \end{gathered}$$

$\mathrm{n}\left(\mathrm{B}\right)=8$

$\mathrm{P}\left(\mathrm{B}\right)=\frac{\mathrm{n}\left(\mathrm{B}\right)}{\mathrm{n}\left(\mathrm{S}\right)}=\frac{8}{25}$

Hence, the probability that both will visit the shop on consecutive days is $\frac{\mathbf{8}}{\mathbf{25}}$.

(iii) Calculating the probability that both will visit the shop on different days:

$\mathrm{P}\left(\mathrm{A}\right)$ be the probability of both will visit the shop on the same day.

Consider $\mathrm{C}$ being an event that both will visit the shop on different days.

$\mathrm{P}\left(\mathrm{C}\right)$ is the probability that both will visit the shop on different days.

$$\begin{gathered} \mathrm{P}\left(\mathrm{C}\right)=1-\mathrm{P}\left(\mathrm{A}\right) \\ \Rightarrow \mathrm{P}\left(\mathrm{C}\right)=1-\frac{1}{5} \\ \Rightarrow \mathrm{P}\left(\mathrm{C}\right)=\frac{4}{5} \end{gathered}$$

Hence, the probability of both will visit the shop on different days is $\frac{\mathbf{4}}{\mathbf{5}}$.