Binomial: Advanced?

The unemployment rate in a city is 13%. Find the probability that at least 2 out of 10 people from this city sampled at random are unemployed. Round your answer to four decimal places.

Binomial: Advanced?

Let X be the number of unemployed sampled. X has the binomial distribution with n = 10 trials and success probability p = 0.13

In general, if X has the binomial distribution with n trials and a success probability of p then

P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)

for values of x = 0, 1, 2, ..., n

P[X = x] = 0 for any other value of x.

The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.

Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.

X ~ Binomial( n , p )

the mean of the binomial distribution is n * p = 1.3

the variance of the binomial distribution is n * p * (1 - p) = 1.131

the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 1.063485

The Probability Mass Function, PMF,

f(X) = P(X = x) is:

P( X = 0 ) = 0.2484234

P( X = 1 ) = 0.3712074

P( X = 2 ) = 0.249605

P( X = 3 ) = 0.09945945

P( X = 4 ) = 0.02600808

P( X = 5 ) = 0.004663517

P( X = 6 ) = 0.0005807061

P( X = 7 ) = 4.95841e-05

P( X = 8 ) = 2.778420e-06

P( X = 9 ) = 9.225914e-08

P( X = 10 ) = 1.378585e-09

The Cumulative Distribution Function, CDF,

F(X) = P(X ≤ x) is:

x

∑ P(X = t) =

t = 0

P( X ≤ 0 ) = 0.2484234

P( X ≤ 1 ) = 0.6196308

P( X ≤ 2 ) = 0.8692358

P( X ≤ 3 ) = 0.9686952

P( X ≤ 4 ) = 0.9947033

P( X ≤ 5 ) = 0.9993668

P( X ≤ 6 ) = 0.9999475

P( X ≤ 7 ) = 0.9999971

P( X ≤ 8 ) ≈ 1

P( X ≤ 9 ) ≈ 1

P( X ≤ 10 ) = 1

1 - F(X) is:

n

∑ P(X = t) =

t = x

P( X ≥ 0 ) = 1

P( X ≥ 1 ) = 0.7515766

P( X ≥ 2 ) = 0.3803692 ← answer = 1 - P(X = 1) - P(X = 0)

P( X ≥ 3 ) = 0.1307642

P( X ≥ 4 ) = 0.03130475

P( X ≥ 5 ) = 0.005296679

P( X ≥ 6 ) = 0.0006331623

P( X ≥ 7 ) = 5.245616e-05

P( X ≥ 8 ) = 2.872057e-06

P( X ≥ 9 ) = 9.363773e-08

P( X ≥ 10 ) = 1.378584e-09

Binomial: Advanced?

loan

The negation of the at least 2 out of 10 people are unemployed is less than two people are unemployed.

Using: 1 - P(A%26#039;) = P(A), you can use this relationship to find your answer.

To find the probability that less than 2 people are unemployed(i.e. 9 or more are employed), you only need to do two combinations.

P(A%26#039;) = (10 C 1)(.87^9)(0.13^1) + (10 C 0)(0.87^10)(0.13^0) ~= 0.6196

Your answer should be:

1 - P(A%26#039;)

1 - P(A%26#039;) = 0.3804