Answer: [tex]0.416655[/tex]
Step-by-step explanation:
If random variable X has the binomial distribution B(n, p), then
[tex]P(X=x)=\ ^nC_xp^x(1-p)^{n-x}[/tex]
, where n= Total number of trials, x = number of successes , p= probability of success in each trial.
Given: The random variable R has the binomial distribution B (12,0.35).
n= 12, = 0.35
[tex]P(R>4)=1-P(R\leq4)\\\\=1-[P(R=0)+P(R=1)+P(R=3)+P(R=4)]\\\\\\=1-[^{12}C_0(0.35)^{0}(1-0.35)^{12}+^{12}C_1(0.35)^{1}(1-0.35)^{11}+^{12}C_2(0.35)^{2}(1-0.35)^{10}+^{12}C_3(0.35)^{3}(1-0.35)^{9}+^{12}C_4(0.35)^{4}(1-0.35)^{8}]\\\\= 1-[(1)(0.65)^{12}+(12)(0.35)(0.65)^{11}+\dfrac{12!}{2!10!}(0.35)^2(0.65)^{10}+\dfrac{12!}{9!3!}(0.35)^3(0.65)^{9}+\dfrac{12!}{8!4!}(0.35)^4(0.65)^{8}]\\\\=1-0.583345\\\\\approx0.416655[/tex]
Hence, P(R>4)=0.416655
