Question

If x is a binomial random​ variable, use the binomial probability table to find the probabilities below.

a. P(x=3) for n=10, p=0.5 b. P(x≤4) for n=15, p=0.3 c. P(x>1) for n=5, p=0.2 d. P(x<6) for n=15, p=0.8 e. P(x≥14) for n=25, p=0.8 f. P(x=3) for n=20, p=0.1

Answer 1

X ~ Binomial(n,p)

Where binomial probability distribution is

P(X) = ⁿC_x p^x (1-p)^n-x

a)

P(X=3) = ¹⁰C₃ 0.5³ 0.5⁷

= 0.1172

b)

P( X <= 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)

= ¹⁵C₀ 0.3⁰ 0.7¹⁵ + ¹⁵C₁ 0.3¹ 0.7¹⁴ + ¹⁵C₂ 0.3¹ 0.7¹³  + ¹⁵C₃ 0.3³ 0.7¹²  + ¹⁵C₄ 0.3⁴ 0.7¹¹

= 0.5155

c)

P(X > 1) = 1 - P( X <=1)

= 1 - { P(X =0) + P(X=1) }

= 1 - ( ⁵C₀ 0.2⁰ 0.7⁵ +  ⁵C₁ 0.2¹ 0.7⁴ )

= 1 - 0.7373

= 0.2627

d)

P( X < 6) = P ( X <= 5)

= P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)

= ¹⁵C₀ 0.8⁰ 0.2¹⁵ +¹⁵C₁ 0.8¹ 0.2¹⁴ +¹⁵C₂ 0.8² 0.2¹³ +¹⁵C₃ 0.8³ 0.2¹² +¹⁵C₄ 0.84 0.2¹¹ +¹⁵C₅ 0.8⁵ 0.2¹⁰

= 0.0001

e)

P(X >= 14) = P(X=14) +P(X=15) +P(X=16) +P(X=17) +P(X=18) +P(X=19) +P(X=20) +P(X=21) +P(X=22) +

P(X=23) +P(X=24) +P(X=25)

= 0.9985

f)

P(X =3) = ²⁰C₃ 0.1³ 0.9¹⁷

= 0.1901