Bayesian Updating and Confirmation Bias in Weather Prediction

1. The probability of having 5 straight cold days in September, according to Aaron's initial belief, is (0.1)^5 = 0.00001, or 0.001%.

2. Let's denote:

• S: The event that it is spring in September.
• W: The event that it is warm.
• C: The event that it is cold.

Aaron initially believes P(S) = 0.9, P(W|S) = 0.9, P(C|S) = 0.1. Aaron also believes P(S') = 0.1, P(W|S') = 0.3, P(C|S') = 0.7.

After observing cold, warm, cold, cold, warm, Aaron needs to update his beliefs using Bayes' theorem.

P(S|CCCWW) = P(S) * P(CCCWW|S) / P(CCCWW) P(S|CCCWW) = P(S) * P(C|S) * P(C|S) * P(C|S) * P(W|S) * P(W|S) / P(CCCWW) P(S|CCCWW) = 0.9 * 0.1 * 0.1 * 0.1 * 0.9 * 0.9 / P(CCCWW) P(S|CCCWW) = 0.0000729 / P(CCCWW)

To find P(CCCWW), we need to consider both cases: September being spring (S) and September being winter (S').

P(CCCWW) = P(CCCWW|S) * P(S) + P(CCCWW|S') * P(S') P(CCCWW) = P(C|S) * P(C|S) * P(C|S) * P(W|S) * P(W|S) * P(S) + P(C|S') * P(C|S') * P(C|S') * P(W|S') * P(W|S') * P(S') P(CCCWW) = 0.1 * 0.1 * 0.1 * 0.9 * 0.9 * 0.9 * 0.9 + 0.7 * 0.7 * 0.7 * 0.3 * 0.3 * 0.1 P(CCCWW) = 0.0000729 + 0.0124743 P(CCCWW) = 0.0125472

Substituting this value back into the equation for P(S|CCCWW):

P(S|CCCWW) = 0.0000729 / 0.0125472 P(S|CCCWW) ≈ 0.0058, or 0.58%.

So, after observing those 5 days, Aaron now believes there is approximately a 0.58% chance that it is spring in September.

3. Let's denote:

• T: The event that the weather forecast says it will be cold.
• F: The event that the weather forecast says it will be warm.

Aaron currently believes P(W) = 0.7 and P(C) = 0.3.

Now, Aaron checks the weather forecast and it says it will be cold tomorrow. Suppose that the forecast is correct 90% of the time.

P(T|W) = 0.9 (probability of a correct cold forecast given that it will be warm) P(F|W) = 1 - P(T|W) = 1 - 0.9 = 0.1 (probability of a correct warm forecast given that it will be warm) P(T|C) = 0.9 (probability of a correct cold forecast given that it will be cold) P(F|C) = 1 - P(T|C) = 1 - 0.9 = 0.1 (probability of a correct warm forecast given that it will be cold)

Suppose Aaron interprets a cold forecast as a warm forecast with a 20% chance (due to his confirmation bias). Let's denote:

• W': The event that Aaron interprets a cold forecast as a warm forecast.
• C': The event that Aaron interprets a warm forecast as a cold forecast.

P(W'|T) = 0.2 (probability of interpreting a cold forecast as a warm forecast given that it is actually cold) P(C'|F) = 0 (probability of interpreting a warm forecast as a cold forecast given that it is actually warm)

We need to update Aaron's belief about the probability that it is going to be warm tomorrow, given his interpretation of a warm forecast.

P(W|T) = P(W) * P(T|W) / P(T) P(W|T) = 0.7 * 0.9 / (P(T|W) * P(W) + P(T|C) * P(C) + P(T|W') * P(W') + P(T|C') * P(C')) P(W|T) = 0.7 * 0.9 / (0.9 * 0.7 + 0.9 * 0.3 + 0.2 * 0.7 + 0 * 0.3) P(W|T) = 0.63 / (0.63 + 0.27 + 0.14 + 0) P(W|T) = 0.63 / 1.04 P(W|T) ≈ 0.6058, or 60.58%.

Therefore, after interpreting a cold forecast as a warm forecast, Aaron's updated belief about the probability that it is going to be warm tomorrow is approximately 60.58%.