question_answer

Two friends visit a restaurant randomly during 5 pm to 6 pm. Among the two, whoever comes first waits for 15 min. and then leaves. The probability that they meet is:

A) \[\frac{1}{4}\]

B) \[\frac{1}{16}\]

C) \[\frac{7}{16}\]

D) \[\frac{9}{16}\]

Correct Answer: C

Solution :

Let the friends come to the restaurant at 5h x min. and 5h y min., respectively, where \[x,y\in [0,60].\] Hence, the sample space consists of all points (x, y) lying in \[60\times 60\]square as shown above and for favourable cases, \[\left| x-y\text{ } \right|\le 15,\] that is \[-15\le x-y\le 15\] which is shown by shaded region in the graph shown below:

Hence, the probability that they will meet \[=1-\frac{2\times \frac{1}{2}\times 45\times 45}{60\times 60}=1-{{\left( \frac{3}{4} \right)}^{2}}=\frac{7}{16}\]