Question

Choose a number n uniformly at random from the set \[\left\{ 1,2,....,100 \right\}\]. Choose one of the first seven days of the year 2014 at random and consider n consecutive days starting with the chosen day. What is the probability that among the chosen n days, either a Monday or Sunday will be chosen?
(a) $\dfrac{1}{7}$,
(b) $\dfrac{2}{7}$,
(c) $\dfrac{12}{49}$,
(d) $\dfrac{3}{175}$.

Answer 1

Hint: We first recall the total number of days present in a week and then find the total numbers that we can get consecutive days by choosing the value of n from the given set of numbers. We then find the values of n that the required condition is not satisfied. We then find the total number of favorable cases in the remaining values of n. We then divide the favorable cases with the total no. of ways to get the required probability.

Complete step-by-step solution
According to the problem, we need to choose a number n from the set \[\left\{ 1,2,....,100 \right\}\]. Now, we need to consider one of the first 7 seven days of the year 2014 at random and write n consecutive days starting with the chosen day. We need to find the probability that there is either a Monday or Sunday present among the written n consecutive days.
We know that every week consists of 7 days namely Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday. Whatever the number we assume for the value n, the number of ways that we can write the n consecutive days without repeating the first day will be 7 for each value of n.
So, the total no. of ways we can write consecutive days is $\left( 100\times 7 \right)=700$ ways.
If we are writing the n consecutive days for the value of n greater than or equal to 7 $(n\text{ }\ge \text{ }7)$. We get both Sunday and Monday on those consecutive days and we get the possible cases only for the values of n less than 7 $\left( n < 7 \right)$.
The total no. of cases will be the same for each value of n less than 7. So, we check for one value of n which will be less than 7, and multiply with 6 to get all the likely cases using it.
Let us consider $n=5$ and let us start with each day of the week and write the 5 consecutive days as shown below.
Case (1): Monday – Tuesday – Wednesday – Thursday – Friday
It is a favorable case because Monday also included.
Case (2): Tuesday – Wednesday – Thursday – Friday – Saturday
It is an unfavorable case as Monday is not included.
Case (3): Wednesday – Thursday – Friday – Saturday – Sunday
It is a favorable case as Sunday is also included.
Case (4): Thursday – Friday – Saturday – Sunday – Monday
It is an unfavorable case as it included both Sunday and Monday, but to get the favorable case
either Monday or Sunday is to be included but not both.
Case (5): Friday – Saturday – Sunday – Monday – Tuesday
It is also an unfavorable case as it included both Sunday and Monday, but to get the favorable case.
either Monday or Sunday is to be included but not both.
Case (6): Saturday – Sunday – Monday – Tuesday – Wednesday
We can see that this is also an unfavorable case as it included both Sunday and Monday, but to get the favorable case.
Case (7): Sunday – Monday – Tuesday – Wednesday – Thursday
We can see that this is also an unfavorable case as it included both Sunday and Monday, but to get the favorable case
Therefore, we got only two favorable cases for $n=5$.
Similarly, we get 2 favourable cases for $n=1,2,3,4,6$.
So, the total number of favorable cases is $\left( 6\times 2 \right)=12$ ways.
$P=\dfrac{\text{Total number of favourable cases}}{\text{Total number of possible cases}}$.
$P=\dfrac{\text{12}}{\text{700}}$.
$P=\dfrac{\text{3}}{\text{175}}$
Hence, the correct option is (d)

Note: We may make mistakes thinking either Monday or Sunday can include both of them, which should always be avoided. Here we should choose the number from the given set first as we cannot assume the number ourselves. We should not make calculation mistakes while solving this problem. We should for $n=1$, we can only 1 day which will be a total consecutive day. We need to make sure that we have checked all the possible while solving the probability problems.