Questions: A poll found that 30% of adults do not work at all while on summer vacation. In a random sample of 6 adults, let x represent the number who do not work during summer vacation. Find the probability that 2 or fewer of the 6 adults do not work during summer vacation. P(x ≤ 2)= (Round to four decimal places as needed.)

Solution Steps

We are given that \(30\%\) of adults do not work during summer vacation. In a random sample of \(n = 6\) adults, we let \(x\) represent the number of adults who do not work during summer vacation. We need to find the probability that \(2\) or fewer of the \(6\) adults do not work, expressed as \(P(x \leq 2)\).

The scenario follows a binomial distribution where:

• The number of trials \(n = 6\),
• The probability of success (an adult not working) \(p = 0.30\),
• The number of successes we are interested in is \(k = 2\).

To find \(P(x \leq 2)\), we compute the cumulative distribution function (CDF) for the binomial distribution with the parameters defined above. This gives us the probability of having \(2\) or fewer successes in \(6\) trials.

After performing the calculation, we find that the cumulative probability is: \[ P(x \leq 2) = 0.7443 \]

This result indicates that there is a \(74.43\%\) chance that \(2\) or fewer of the \(6\) adults do not work during summer vacation.

Final Answer