This problem involves conditional probability, which is the probability of an event occurring given that another event has already occurred. We need to determine the probability that it has not rained on a day, given that a village fair did not make a loss on that day during the rainy season. This can be effectively solved using Bayes' Theorem.

Understanding the Given Probabilities

Let's define the events involved in this probability problem:

• Let R be the event that it rains on a given day.
• Let R' be the event that it does not rain on a given day.
• Let L be the event that the village fair makes a loss.
• Let L' be the event that the village fair does not make a loss (i.e., it makes a profit or breaks even).

From the problem statement, we are provided with the following probabilities:

• The probability of rain, P(R) = 70% = 0.70.
• Consequently, the probability of no rain, P(R') = 1 - P(R) = 1 - 0.70 = 0.30.
• The conditional probability of the fair making a loss if it rains, P(L | R) = 80% = 0.80.
• The conditional probability of the fair making a loss if it does not rain, P(L | R') = 10% = 0.10.

Calculating Required Probabilities

To use Bayes' Theorem, we first need to find the probabilities of the fair not making a loss under both conditions (rain and no rain). These are the complements of the given conditional probabilities of making a loss.

• Probability of not making a loss if it rains (P(L' | R)):

  If the chance of making a loss when it rains is 80%, then the chance of not making a loss is \(1 - 0.80\).

  \(P(L' | R) = 1 - P(L | R) = 1 - 0.80 = 0.20\)

• Probability of not making a loss if it does not rain (P(L' | R')):

  If the chance of making a loss when it does not rain is 10%, then the chance of not making a loss is \(1 - 0.10\).

  \(P(L' | R') = 1 - P(L | R') = 1 - 0.10 = 0.90\)

Let's summarize the derived probabilities in a table for clarity:

Calculating the Total Probability of No Loss

Next, we need to find the overall probability that the fair does not make a loss on a given day, P(L'). We can do this using the law of total probability, which sums the probabilities of not making a loss under all possible conditions (rain or no rain).

The formula for the total probability of an event \(L'\) is:

\(P(L') = P(L' | R) \times P(R) + P(L' | R') \times P(R')\)

Substitute the values we have:

\(P(L') = (0.20 \times 0.70) + (0.90 \times 0.30)\)

\(P(L') = 0.14 + 0.27\)

\(P(L') = 0.41\)

Applying Bayes' Theorem for No Rain Probability

We are asked to find the probability that it has not rained on that day, given that the fair has not made a loss. This is \(P(R' | L')\). Bayes' Theorem states:

\(P(R' | L') = \frac{P(L' | R') \times P(R')}{P(L')}\)

Now, substitute the values we have calculated:

• \(P(L' | R') = 0.90\)
• \(P(R') = 0.30\)
• \(P(L') = 0.41\)

\(P(R' | L') = \frac{0.90 \times 0.30}{0.41}\)

\(P(R' | L') = \frac{0.27}{0.41}\)

Final Probability Result

The probability that it has not rained on a day, given that the fair has not made a loss, is \(27/41\).