What is the probability that a randomly taken leap year has 53 days?

Answer

Verified

Hint: There are $366$ days in leap year means $52$ weeks and $2$ extra days. Make the possibilities for two extra days and evaluate the probability.Probability of any event A is defined as the ratio of the favourable outcomes to the total outcomes. The formula for the probability of A will be:\[P(A) = \dfrac{{Favourable\,Outcomes}}{{Total\,Outcomes}}\]

Complete step-by-step answer:

We have given a leap year.We have to evaluate the probability that a leap year has $53$ Sundays.The difference between the leap and normal year is that the number of days in normal year is $365$ and the number of days in leap year is $366$Therefore, in the leap year there are $52$ weeks and $2$ extra days. It means $52$ Sundays are included.We have to make the conditions for $2$ extra days and our favourable outcomes will consist of $1$ Sunday so that total Sundays will be $53$.The possibilities for two extra days will be:{Monday, Tuesday}, {Tuesday, Wednesday}, {Wednesday, Thursday}, {Thursday, Friday}, {Friday, Saturday}, {Saturday, Sunday} and {Sunday, Monday}In two of the cases {Saturday, Sunday} and {Sunday, Monday}, Sunday is present, therefore favourable outcomes will be $2$ and total possibilities are $7$, therefore, total outcomes will be $7$.We know that probability of any event A is defined as the ratio of the favourable outcomes to the total outcomes. The formula for the probability of A will be:\[P(A) = \dfrac{{Favourable\,Outcomes}}{{Total\,Outcomes}}\]Therefore, the probability of $53$ Sundays in a leap year is $\dfrac{2}{7}$.

So, the correct answer is “Option B”.

Note: In these types of questions the total outcomes will not be equal to the total number of days because in $365$ days, the number of Sundays are fixed. Therefore, the total outcomes will come from $2$ extra days.

Find the probability that a leap year selected at random will contain 53 Sundays.

A leap year has 366 days with 52 weeks and 2 days.

Now, 52 weeks conatins 52 sundays.

The remaining two days can be:

Sunday and Monday
Monday and Tuesday
Tuesday and Wednesday
Wednesday and Thursday
Thursday and Friday
Friday and Saturday
Saturday and Sunday

Out of these 7 cases, there are two cases favouring it to be Sunday.

∴ P(a leap year having 53 Sundays) = `("Number of favourable outcomes")/"Number of all possible outcomes"`

`= 2/7`

Thus, the probability that a leap year selected at random will contain 53 Sundays is `2/7`.

Concept: Concept Or Properties of Probability

Is there an error in this question or solution?

Text Solution

`(7)/(366)``(28)/(183)``(1)/(7)``(2)/(7)`

Answer : D

Solution : A leap year has 366 days in which 52 and 2 days are extra. i.e. (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thrusdar), (Thursday, Friday), (Friday, Saturday), (Saturday, Sunday). So, probability that a leap year contains 53 Sunday = 2/7.

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