Respuesta :
Answer:
(1) The probability of the event X = 2 is 0.0413.
(2) The probability of the event X ≤ 1 is 0.009.
(3) The probability of the event X < 6 is 0.6846.
Step-by-step explanation:
Let the random variable X follow a Binomial distribution with parameter n = 8 and p = 0.60.
The probability mass function of the Binomial distribution is:
[tex]P(X=x)={n\choose x}p^{x}(1-p)^{n-x};\ x=0, 1, 2,...[/tex]
(1)
Compute the probability of the event X = 2 as follows:
[tex]P(X=2)={8\choose 2}(0.60)^{2}(1-0.60)^{8-2}\\=\frac{8!}{2!(8-2)!}\times (0.60)^{2}\times(0.40)^{6}\\=28\times0.36\times0.004096\\=0.0413[/tex]
Thus, the probability of the event X = 2 is 0.0413.
(2)
Compute the probability of the event X ≤ 1 as follows:
P (X ≤ 1) = P (X = 0) + P (X = 1)
[tex]={8\choose 0}(0.60)^{0}(1-0.60)^{8-0}+{8\choose 1}(0.60)^{1}(1-0.60)^{8-1}\\=\frac{8!}{0!(8-0)!}\times (0.60)^{0}\times(0.40)^{8}+\frac{8!}{1!(8-1)!}\times (0.60)^{1}\times(0.40)^{7}\\=(1\times1\times0.00066)+(8\times0.60\times0.00164)\\=0.008532\approx0.009[/tex]
Thus, the probability of the event X ≤ 1 is 0.009.
(3)
Compute the probability of the event X < 6 as follows:
P (X < 6) = 1 - P (X ≥ 6)
= 1 - P (X = 6) - P (X = 7) - P (X = 8)
[tex]=1-{8\choose 6}(0.60)^{6}(1-0.60)^{8-6}+{8\choose 7}(0.60)^{7}(1-0.60)^{8-7}+{8\choose 8}(0.60)^{8}(1-0.60)^{8-8}\\=1-\frac{8!}{6!(8-6)!}\times (0.60)^{6}\times(0.40)^{2}+\frac{8!}{7!(8-7)!}\times (0.60)^{7}\times(0.40)^{1}\\+\frac{8!}{8!(8-8)!}\times (0.60)^{8}\times(0.40)^{0}\\=1-0.2090-0.0896-0.0168\\=0.6846[/tex]
Thus, the probability of the event X < 6 is 0.6846.
