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Sagot :
Using the binomial distribution, it is found that there is a 0.9842 = 98.42% probability that 3 or fewer experienced insomnia as a side effect, which means that it is a highly likely event.
What is the binomial distribution formula?
The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
The values of the parameters are given as follows:
n = 20, p = 0.05.
The probability that 3 or fewer experienced insomnia as a side effect is given by:
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]
Hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{20,0}.(0.05)^{0}.(0.95)^{20} = 0.3585[/tex]
[tex]P(X = 1) = C_{20,1}.(0.05)^{1}.(0.95)^{19} = 0.3774[/tex]
[tex]P(X = 2) = C_{20,2}.(0.05)^{2}.(0.95)^{18} = 0.1887[/tex]
[tex]P(X = 3) = C_{20,3}.(0.05)^{3}.(0.95)^{17} = 0.0596[/tex]
Then:
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.3585 + 0.3774 + 0.1887 + 0.0596 = 0.9842[/tex]
0.9842 = 98.42% probability that 3 or fewer experienced insomnia as a side effect.
Since this probability is greater than 95%, this is a highly likely event.
More can be learned about the binomial distribution at https://brainly.com/question/24863377
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