Answer:
0.9940
Step-by-step explanation:
P(at least 1) = 1 − P(zero)
P(at least 1) = 1 − (1 − 0.64)⁵
P(at least 1) = 1 − (0.36)⁵
P(at least 1) = 0.9940
The probability that at least one of them has been vaccinated is 0.9939.
The binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either success or failure. It helps to check the probability of getting “x” successes in “n” independent trials of a binomial experiment.
For the given situation,
Number of people vaccinated = 64% = 0.64
The formula of binomial distribution is
[tex]P(x:n,p) = nC_{x} p^x (1-p)^{n-x}[/tex]
Here x is the number of successes, x ≤ 1
n is the number of trials, n = 5
p is the probability of a success on a single trial, p = 0.64 and
where, [tex]nC_{x}=\frac{n!}{x!(n-x)!}[/tex]
The probability is [tex]P(X \leq 1)=1-P(X=0)[/tex]
[tex]P(X=0)= 5C_{0} (0.64)^{0} (1-0.64)^{5-0}[/tex]
⇒ [tex]P(X=0)= 1(1) (0.36)^{5}[/tex]
⇒ [tex]P(X=0)= 0.0060[/tex]
Thus, [tex]P(X \leq 1)=1-P(X=0)\\[/tex]
⇒ [tex]P(X \leq 1)=1-0.0060[/tex]
⇒ [tex]P(X \leq 1)=0.9939[/tex]
Hence we can conclude that the probability that at least one of them has been vaccinated is 0.9939.
Learn more about binomial distribution here
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