A survey shows that $$ 63 \% $$ of the Americans like cheese whereas $$ 76 \% $$ like apples. If $$ x \% $$ of the Americans like both cheese and apples, then $$ \begin{array}{ll} ext { (a) } x=39 & ext { (b) } x=63 \ ext { (c) } 39 \leq x \leq 63 & ext { (d) None of these }\end{array} $$

Question

A survey shows that $$ 63 \% $$ of the Americans like cheese whereas $$ 76 \% $$ like apples. If $$ x \% $$ of the Americans like both cheese and apples, then $$ \begin{array}{ll}	ext { (a) } x=39 & 	ext { (b) } x=63 \ 	ext { (c) } 39 \leq x \leq 63 & 	ext { (d) None of these }\end{array} $$

UpStudy AI · Accepted Answer

To determine the possible values of $$ x \% $$ (the percentage of Americans who like both cheese and apples), we can use the principle of **inclusion-exclusion** in probability.

Given:
- $$ P(\text{Cheese}) = 63\% $$
- $$ P(\text{Apples}) = 76\% $$

The formula for the probability of the union of two events is:
$$
P(\text{Cheese} \cup \text{Apples}) = P(\text{Cheese}) + P(\text{Apples}) - P(\text{Cheese} \cap \text{Apples})
$$
However, since the probability cannot exceed 100%, we can derive the bounds for $$ x \% $$ as follows:

1. **Minimum Value of $$ x \% $$:**
   $$
   x\% \geq P(\text{Cheese}) + P(\text{Apples}) - 100\% = 63\% + 76\% - 100\% = 39\%
   $$

2. **Maximum Value of $$ x \% $$:**
   Since $$ x \% $$ cannot exceed the smaller of the two individual probabilities:
   $$
   x\% \leq \min(63\%, 76\%) = 63\%
   $$

Therefore, the percentage of Americans who like both cheese and apples must satisfy:
$$
39\% \leq x \% \leq 63\%
$$

**Answer:**

Problem (c) is correct. The percentage x satisfies 39 ≤ x ≤ 63.
The percentage of Americans who like both cheese and apples is between 39% and 63%.

A survey shows that of the Americans like cheese whereas
like apples. If of the Americans like both cheese and
apples, then

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