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If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?

Last updated date: 22nd Jun 2024
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Hint: In order to find the probability that the leap year has 53 Tuesdays, we calculate the number of weeks in a leap year and the number of remaining days that do not make up a whole week. The probability of having 53 Tuesdays in a leap year is nothing but the probability of occurrence of a Tuesday in these extra days. Use the formula of probability.

Complete step-by-step answer:
Given Data,
A leap year.

We know a leap year consists of 366 days.
That makes up for$\dfrac{{366}}{7}$= 52 weeks + 2 days.
It is obvious that there are 52 Tuesdays in those 52 weeks, therefore the probability that the leap year has 53 Tuesdays is the probability that the remaining two days will be Tuesdays.

The remaining two days can be any two consecutive days of the week:
Monday and Tuesday
Tuesday and Wednesday
Wednesday and Thursday
Thursday and Friday
Friday and Saturday
Saturday and Sunday
Sunday and Monday.
Which are a total of 7 possible outcomes for those two extra days.
The possible outcomes of these two days containing a Tuesday is 2.

We know the formula of probability is defined as,${\text{P = }}\dfrac{{{\text{favorable outcomes}}}}{{{\text{total outcomes}}}}$
Therefore the probability that the given random leap year has 53 Tuesdays is given by,
${\text{P}}\left( {\text{T}} \right){\text{ = }}\dfrac{2}{7} = 0.28$

Note:In order to solve this type of problems the key is to know the concept of a leap year. A leap year has 366 days unlike a regular year which has 365 days. Dividing the 366 days into weeks and extra days is a vital step in solving this problem, it breaks down the problem to the probability of occurrence of Tuesday in the extra days. Once we understand this logic, we list out the possibilities of all the outcomes and substitute them in the formula of probability.