Question

When leaving her house, Pam takes her umbrella with her on 90% of rainy days, and only in 20% of non-rainy days. It is known that Pam's town gets rain on 70% of the days. On a given day, if we see Pam outside with her umbrella, what is the probability that it is raining in her town?

Answer 1

Using Bayes' Theorem and the provided probabilities, we calculate that if Pam is seen with an umbrella, the probability that it is raining in her town is approximately 91.3%.

Step-by-step explanation:

To calculate the probability that it is raining in Pam's town given that she is seen with an umbrella, we can use Bayes' Theorem. Let R denote the event that it is raining and U denote the event that Pam carries an umbrella. We are given P(U|R) = 0.9, P(U|¬R) = 0.2, and P(R) = 0.7, where ¬R denotes the event that it is not raining. We want to find P(R|U), the probability it is raining given that Pam has her umbrella.

We start by finding P(U), the probability that Pam has her umbrella:

• 
  
• P(U) = P(U|R)P(R) + P(U|¬R)P(¬R)
• 
  
• P(U) = (0.9)(0.7) + (0.2)(0.3)
• 
  
• P(U) = 0.63 + 0.06
• 
  
• P(U) = 0.69

Now we can apply Bayes' Theorem to find P(R|U):

• 
  
• P(R|U) = P(U|R)P(R) / P(U)
• 
  
• P(R|U) = (0.9)(0.7) / 0.69
• 
  
• P(R|U) = 0.63 / 0.69
• 
  
• P(R|U) ≈ 0.913

So, if Pam is seen with an umbrella, the probability that it is raining is approximately 91.3%.