. A survey shows that 76% of the Indians like Oranges, whereas 38% like both oranges and bananas. What percentage of Indians like Bananas?

1. Break Down the Problem

We need to determine the percentage of Indians who like bananas, given:

• $P(A) = 76\%$ (percentage of Indians who like oranges)
• $P(A \cap B) = 38\%$ (percentage of Indians who like both oranges and bananas)

We want to find $P(B)$, the percentage of Indians who like bananas.

2. Relevant Concepts

We can utilize the principle of inclusion-exclusion in probability, which states: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ where:

• $P(A \cup B)$ is the probability that a person likes either oranges or bananas or both.

However, in this scenario, we do not have $P(A \cup B)$ directly. Instead, we can express $P(B)$ using: $P(B) = P(A \cap B) + P(B \setminus A)$ but we will use the total union of interests to find $P(B)$.

3. Analysis and Detail

Using the known information:

1. Let $P(A) = 0.76$ (as a decimal)
2. Let $P(A \cap B) = 0.38$ (as a decimal)

Now assume: $P(B) = P(A \cap B) + P(A^c \cap B)$ where $P(A^c$ is the complement of $P(A)$ (non-orange likers).

We want to express this to find $P(B)$.

We know: $P(B) = P(A \cap B) + \text{(Percentage liking only Bananas)}$ From the equation for $P(B)$: $P(B) = P(A \cap B) + (1 - P(A) - P(B \cap A^c))$ This is complex to solve without further data, but we shall consider only $P(B \cap A)$.

4. Verify and Summarize

To find the percentage of Indians who like only bananas, further specific data is needed. However, assuming that the proportions of those surveyed are in aggregate, we can initially assume total likability as a standard among the surveyed:

Taking a simplified assumption, If both propositions are as we know, we can summarize that: $P(B) \geq P(A \cap B) \text{ since it is common between both.}$

Assuming bananas have a non-overlapping category, we can take: $P(B) \leq 100\% - P(A)$ We cannot conclude directly without indicating specific surveys of specific liking.

Since we can't determine an exact value for $P(B)$ without further data, we note:

• Indians who like bananas at minimum equal $P(A \cap B)$.

Final Answer

The percentage of Indians who like bananas cannot be distinctly calculated without additional data indicating unique likers. Hence, it remains $P(B) \geq 38\%$ (as they cannot exceed not liking bananas at all).