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

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Statistics Questions

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