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प्रश्न
What is the probability that a leap year has 53 Tuesdays and 53 Mondays?
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उत्तर
Given: A leap year contains 366 days.
\[ 366\ \text{days} = 52\ \text{weeks} + 2\ \text{extra days} \]
52 weeks contain 52 Mondays and 52 Tuesdays for certain.
The remaining 2 extra consecutive days can be any of the following 7 pairs:
\[ S = {\text{(Sunday, Monday)}, \text{(Monday, Tuesday)}, \text{(Tuesday, Wednesday)}, \text{(Wednesday, Thursday)}, \text{(Thursday, Friday)}, \text{(Friday, Saturday)}, \text{(Saturday, Sunday)}} \]
\[ n(S) = 7 \]
For the year to have both 53 Mondays and 53 Tuesdays, the two extra days must be Monday and Tuesday.
Favourable outcome = `({\text{(Monday, Tuesday)}})`
\[ m = 1 \]
Answer: \[ P(53\ \text{Mondays and } 53\ \text{Tuesdays}) = \dfrac{m}{n(S)} = \dfrac{1}{7} \]
