Let's first transcribe the text from the image and break down the problem to solve it:

The text in the image appears to be in Spanish and states:

"4. Un cliente recibe una encuesta en una plaza comercial muy concurrida. La siguiente tabla proporciona información sobre el tipo de café seleccionado por alguien que compra una taza de café en un lunes de octubre del año pasado:

Considera la posibilidad de seleccionar al azar a un comprador de café.

a. ¿Cuál es la probabilidad de que la persona adopte una taza pequeña? ¿Y una taza de café descafeinado?

b. Si nos enteramos de que la persona seleccionada compró una taza de café pequeña, ¿cuál es la probabilidad de que compre café descafeinado y cómo se interpreta esta probabilidad?"

Solution:

Part (a):

We are asked for two probabilities:

1. Probability of choosing a small ("pequeña") cup: To find the probability that the person selects a small cup, we sum all the probabilities associated with the small cup size (both regular and decaffeinated). From the table:

   • Probability of small regular: $10\% = 0.10$
   • Probability of small decaffeinated: $10\% = 0.10$

   Therefore, the total probability of choosing a small cup is: $P(\text{Small}) = 0.10 + 0.10 = 0.20$ So, the probability is 0.20 or 20%.

2. Probability of choosing decaffeinated coffee: To find the probability of selecting a decaffeinated coffee, we sum the probabilities for all cup sizes that are decaffeinated:

   • Probability of small decaffeinated: $10\% = 0.10$
   • Probability of medium decaffeinated: $10\% = 0.10$
   • Probability of large decaffeinated: $10\% = 0.10$

   Thus, the total probability of choosing decaffeinated coffee is: $P(\text{Decaffeinated}) = 0.10 + 0.10 + 0.10 = 0.30$ So, the probability is 0.30 or 30%.

Part (b):

This is a conditional probability problem, where we are given that the person has selected a small cup, and we need to find the probability that the coffee is decaffeinated.

We are looking for $P(\text{Decaffeinated | Small})$, which is the conditional probability that the coffee is decaffeinated given that it is a small cup.

We use the formula for conditional probability: $P(\text{Decaffeinated | Small}) = \frac{P(\text{Decaffeinated and Small})}{P(\text{Small})}$

From the table:

• $P(\text{Decaffeinated and Small}) = 0.10$ (from the decaffeinated small category).
• $P(\text{Small}) = 0.20$ (calculated in part (a)).

Thus, the conditional probability is: $P(\text{Decaffeinated | Small}) = \frac{0.10}{0.20} = 0.50$ So, the probability is 0.50 or 50%.

Interpretation:

This means that if we know the person has chosen a small coffee, there is a 50% chance that the coffee is decaffeinated.

Would you like more details or have any questions? Here are five related questions:

1. What is the probability of selecting a large coffee?
2. How would the conditional probability change if we wanted the probability of selecting regular coffee given a large cup?
3. How do you interpret conditional probabilities in real-life scenarios?
4. Can we apply Bayes' theorem to solve part (b)? How?
5. What is the total probability of selecting either regular or decaffeinated coffee, regardless of size?

Tip: Conditional probabilities are particularly useful in scenarios where additional information narrows down the possibilities of an outcome, like knowing a coffee size before determining if it's decaffeinated.