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Answer:
Step-by-step explanation:
To find the probability that exactly 16 out of 20 randomly selected adults under age 35 have eaten pizza for breakfast, we can use the binomial probability formula.
The situation fits the criteria for a binomial distribution because:
- Each adult either has or has not eaten pizza for breakfast.
- The probability \( p \) of an adult having eaten pizza for breakfast is \( \frac{3}{4} \).
- The number of trials \( n \) is 20 (the sample size).
- We are interested in the probability of exactly 16 adults having eaten pizza for breakfast.
The binomial probability formula is:
\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \]
where:
- \( n \) is the number of trials,
- \( k \) is the number of successes (adults who have eaten pizza for breakfast),
- \( p \) is the probability of success on each trial,
- \( \binom{n}{k} \) is the binomial coefficient.
Here, \( n = 20 \), \( k = 16 \), and \( p = \frac{3}{4} \).
First, calculate \( \binom{20}{16} \):
\[ \binom{20}{16} = \frac{20 \cdot 19 \cdot 18 \cdot 17}{4 \cdot 3 \cdot 2 \cdot 1} = 4845 \]
Next, compute \( p^{16} \) and \( (1-p)^{4} \):
\[ p^{16} = \left(\frac{3}{4}\right)^{16} \]
\[ (1-p)^{4} = \left(\frac{1}{4}\right)^{4} = \frac{1}{256} \]
Now, multiply these together:
\[ P(X = 16) = 4845 \cdot \left(\frac{3}{4}\right)^{16} \cdot \frac{1}{256} \]
Calculate \( \left(\frac{3}{4}\right)^{16} \):
\[ \left(\frac{3}{4}\right)^{16} = \frac{43046721}{4294967296} \]
Multiply by \( \frac{1}{256} \):
\[ \frac{43046721}{4294967296} \cdot \frac{1}{256} = \frac{43046721}{1099511627776} \]
Therefore, the probability that exactly 16 out of 20 adults under age 35 have eaten pizza for breakfast is \( \boxed{\frac{43046721}{1099511627776}} \).
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