Meteorologists from a regional office want to analyze, in a simplified way, the weather behavior in a coastal city to study how it influences whether residents carry umbrellas in the morning. After collecting preliminary data and conducting street interviews, they decided to model the situation with a Hidden Markov Model (HMM) considering whether the day will be sunny, rainy, or cloudy.

On the first day, based on historical city data and the date measurements begin, meteorologists estimate:

• 50% probability of being sunny.

• 30% probability of being rainy.

• 20% probability of being cloudy.

For subsequent days, experts believe the weather retains a "memory" of one day (first-order property), meaning that knowing the weather on one day defines the probabilities for the next day:

• If a day is Sunny:

• 30% of the time it transitions to rainy the next day.

• 10% of the time it transitions to cloudy.

• If a day is Rainy:

• 20% of the time it becomes sunny the next day.

• 30% of the time it transitions to cloudy.

• If a day is Cloudy:

• 30% of the time it clears to sunny the next day.

• 30% of the time it intensifies into rain.

Additionally, a research group conducts surveys each morning to determine how many people carry umbrellas. In a pilot survey, they interviewed 100 people for each type of day (Sunny, Rainy, Cloudy) and obtained the following results:

• Sunny day: 10 out of 100 people carried umbrellas for precaution; the other 90 did not.

• Rainy day: 90 out of 100 people carried umbrellas, and only 10 did not.

• Cloudy day: 60 out of 100 people carried umbrellas, while 40 did not.

These probabilities are considered representative of the observed behavior in this coastal city.