2. In the month of July, 21 days went to the gym and 23 days to work. Find the probability of: a) The days that only work was done b) the days you worked and went to the gym c) The days we went to the gym

a)
Given:
|A| = 21
|B| = 23
|A \cup B| = 31

Using the formula for the union of two sets:
|A \cup B| = |A| + |B| - |A \cap B|

We find:
31 = 21 + 23 - |A \cap B|
|A \cap B| = 44 - 31
|A \cap B| = 13

The number of days where only work was done:
|B| - |A \cap B| = 23 - 13 = 10

The probability of only work being done:
P(\text{only work}) = \frac{10}{31}

Answer: \frac{10}{31}

b)
The number of days where both events occurred:
|A \cap B| = 13

The probability of working and going to the gym:
P(\text{work and gym}) = \frac{13}{31}

Answer: \frac{13}{31}

c)
The number of days where the gym was visited:
|A| = 21

The probability of going to the gym:
P(\text{gym}) = \frac{21}{31}

Answer: \frac{21}{31}