Question

dr. swift, a dermatologist, is creating a face wash for her patients with severe skin allergies. however, the product is not yet ready to be released to the public. tests indicate 43% of people with \highly sensitive skin\ who sampled the facial wash still exhibit an allergic reaction. assume the test results are representative of the general population. if dr. swift released the face wash to the public now, and 5 people with \highly sensitive skin\ used it, what is the probability that at most 1 will have an allergic reaction? write your answer as a decimal rounded to the nearest thousandth. save answer

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Here, \(n = 5\), \(p=0.43\), and we want \(P(X\leq1)=P(X = 0)+P(X = 1)\)

Step2: Calculate \(P(X = 0)\)

For \(k = 0\):
\(C(5,0)=\frac{5!}{0!(5-0)!}=1\)
\(P(X = 0)=C(5,0)\times(0.43)^{0}\times(1 - 0.43)^{5-0}\)
\(=1\times1\times(0.57)^{5}\)
\(=(0.57)^{5}=0.060169257\)

Step3: Calculate \(P(X = 1)\)

For \(k = 1\):
\(C(5,1)=\frac{5!}{1!(5 - 1)!}=\frac{5!}{1!4!}=5\)
\(P(X = 1)=C(5,1)\times(0.43)^{1}\times(1 - 0.43)^{5 - 1}\)
\(=5\times0.43\times(0.57)^{4}\)
\(=5\times0.43\times0.1055593\)
\(=5\times0.0453905\)
\(=0.2269525\)

Step4: Calculate \(P(X\leq1)\)

\(P(X\leq1)=P(X = 0)+P(X = 1)\)
\(=0.060169257+0.2269525\)
\(=0.287121757\approx0.287\)

Answer:

\(0.287\)