The probability that a leap year selected at random will have 53 Mondays is

A leap year has 366 days.

Dividing 366 by 7:

$366 \div 7 = 52$ weeks with remainder $2$

$366 = 52 \times 7 + 2$

$366 = 364 + 2$

A leap year has 52 complete weeks and 2 extra days.

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In 52 complete weeks, every day of the week (Monday, Tuesday, Wednesday, etc.) appears exactly 52 times.

The 2 extra days will each occur 53 times instead of 52 times.

For Monday to occur 53 times, Monday must be one of those 2 extra days.

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The 2 extra days are consecutive days. The possible pairs of consecutive days are:

1. (Sunday, Monday)
2. (Monday, Tuesday)
3. (Tuesday, Wednesday)
4. (Wednesday, Thursday)
5. (Thursday, Friday)
6. (Friday, Saturday)
7. (Saturday, Sunday)

Total possible pairs $= 7$

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Monday appears in 2 of these pairs:

• (Sunday, Monday)
• (Monday, Tuesday)

Favorable outcomes $= 2$

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Probability $= \dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}$

Probability $= \dfrac{2}{7}$

Therefore, the probability that a leap year selected at random will have 53 Mondays is $\dfrac{2}{7}$.