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Consider the following statements:
$1.$ The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year.
$2.$ The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. Which of the statements given above is/are correct?
Options:
$1.$ The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year.
$2.$ The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. Which of the statements given above is/are correct?
Solution:
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Verified Answer
The correct answer is:
1 only
Statement 1:
Anon leap year has 365 days. i.e., 52 weeks and 1 day. 1 day can be \{Sunday\}, \{Monday $\},$ Tuesday\},
\{Wednesday $\}$, \{Thursday\}, \{Friday\}, \{Saturday\}
In total, there are 7 possibilities and 1 possibility is Sunday.
Required probability $=\frac{1}{7}$
A leap year has 366 days. i.e., 52 weeks and 2 days. 2 days can be \{Sun, Mon\}, \{Mon, Tue\}, \{Tue, Wed $\}$,
\{Wed, Thu \}, \{Thu, Fri\}, \{Fri, Sat\}, \{Sat, Sun\}.
In total, there are 7 possibilities and 2 possibilities have Sundays.
Required probability $=\frac{2}{7}$
So, statement 1 is correct. Statement 2 :
March has 31 days. i.e., 4 complete weeks and 3 days. 3 days can be $\{\mathrm{S}, \mathrm{M}, \mathrm{T}\},\{\mathrm{M}, \mathrm{T}, \mathrm{W}\},\{\mathrm{T}, \mathrm{W}, \mathrm{Th}\},\{\mathrm{W}, \mathrm{Th}, \mathrm{F}\}$
$\{\mathrm{Th}, \mathrm{F}, \mathrm{Sa}\},\{\mathrm{F}, \mathrm{Sa}, \mathrm{S}\},\{\mathrm{Sa}, \mathrm{S}, \mathrm{M}\}$
In total 7 possibilities, Monday can come in 3 possibilities
$\therefore$ Required probabilities $=\frac{3}{7}$
April has 30 days. i.e., 4 complete weeks and 2 days. 2 days can be $\{\mathrm{S}, \mathrm{M}\},\{\mathrm{M}, \mathrm{T}\},\{\mathrm{T}, \mathrm{W}\},\{\mathrm{W}, \mathrm{Th}\},\{\mathrm{Th}, \mathrm{F}\}$
$\{\mathrm{F}, \mathrm{Sa}\},\{\mathrm{Sa}, \mathrm{S}\}$
In total 7 possibilities, Monday can come in 2 possibilities.
$\therefore$ Required probability $=\frac{2}{7}$
$\therefore$ Statement 2 is wrong.
Anon leap year has 365 days. i.e., 52 weeks and 1 day. 1 day can be \{Sunday\}, \{Monday $\},$ Tuesday\},
\{Wednesday $\}$, \{Thursday\}, \{Friday\}, \{Saturday\}
In total, there are 7 possibilities and 1 possibility is Sunday.
Required probability $=\frac{1}{7}$
A leap year has 366 days. i.e., 52 weeks and 2 days. 2 days can be \{Sun, Mon\}, \{Mon, Tue\}, \{Tue, Wed $\}$,
\{Wed, Thu \}, \{Thu, Fri\}, \{Fri, Sat\}, \{Sat, Sun\}.
In total, there are 7 possibilities and 2 possibilities have Sundays.
Required probability $=\frac{2}{7}$
So, statement 1 is correct. Statement 2 :
March has 31 days. i.e., 4 complete weeks and 3 days. 3 days can be $\{\mathrm{S}, \mathrm{M}, \mathrm{T}\},\{\mathrm{M}, \mathrm{T}, \mathrm{W}\},\{\mathrm{T}, \mathrm{W}, \mathrm{Th}\},\{\mathrm{W}, \mathrm{Th}, \mathrm{F}\}$
$\{\mathrm{Th}, \mathrm{F}, \mathrm{Sa}\},\{\mathrm{F}, \mathrm{Sa}, \mathrm{S}\},\{\mathrm{Sa}, \mathrm{S}, \mathrm{M}\}$
In total 7 possibilities, Monday can come in 3 possibilities
$\therefore$ Required probabilities $=\frac{3}{7}$
April has 30 days. i.e., 4 complete weeks and 2 days. 2 days can be $\{\mathrm{S}, \mathrm{M}\},\{\mathrm{M}, \mathrm{T}\},\{\mathrm{T}, \mathrm{W}\},\{\mathrm{W}, \mathrm{Th}\},\{\mathrm{Th}, \mathrm{F}\}$
$\{\mathrm{F}, \mathrm{Sa}\},\{\mathrm{Sa}, \mathrm{S}\}$
In total 7 possibilities, Monday can come in 2 possibilities.
$\therefore$ Required probability $=\frac{2}{7}$
$\therefore$ Statement 2 is wrong.
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