A survey showed that $78 \%$ of adults need correction (eyeglasses, contacts, surgery, etc.) for their eyesight. If 15 adults are randomly selected, find the probability that no more than 1 of them need correction for their eyesight. Is 1 a significantly low number of adults requiring eyesight correction?
The probability that no more than 1 of the 15 adults require eyesight correction is (Round to three decimal places as needed.)
Is 1 a significantly low number of adults requiring eyesight correction? Note that a small probability is one that is less than 0.05 .
A. Yes, because the probability of this occurring is small.
B. No, because the probability of this occurring is not small.
C. No, because the probability of this occurring is small.
D. Yes, because the probability of this occurring is not small.
Answer
Final Answer: The probability that no more than 1 out of 15 adults need eyesight correction is approximately \(\boxed{7.416 \times 10^{-9}}\). Therefore, the answer to the question 'Is 1 a significantly low number of adults requiring eyesight correction?' is 'Yes, because the probability of this occurring is small.'
Step 1 ：This is a binomial probability problem. We are given that the probability of success (an adult needing eyesight correction) is 0.78. We are asked to find the probability of 0 or 1 success in 15 trials. We can use the binomial probability formula to solve this problem.
Step 2 ：The binomial probability formula is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: \(P(X=k)\) is the probability of k successes in n trials, \(C(n, k)\) is the number of combinations of n items taken k at a time, p is the probability of success, and n is the number of trials.
Step 3 ：We need to calculate \(P(X=0)\) and \(P(X=1)\), and then add these two probabilities together to get the final answer.
Step 4 ：Given p = 0.78 and n = 15, we calculate \(P_0 = 1.368800680154118e-10\) and \(P_1 = 7.279530889910539e-09\).
Step 5 ：The final probability is the sum of \(P_0\) and \(P_1\), which is \(7.416410957925951e-09\).
Step 6 ：The final probability is extremely small, much less than 0.05. This means that the probability of no more than 1 out of 15 adults needing eyesight correction is very low.
Step 7 ：Final Answer: The probability that no more than 1 out of 15 adults need eyesight correction is approximately \(\boxed{7.416 \times 10^{-9}}\). Therefore, the answer to the question 'Is 1 a significantly low number of adults requiring eyesight correction?' is 'Yes, because the probability of this occurring is small.'
