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Answer the following:

(A) Find the binomial probability P(x = 4), where n = 12 and p = 0.30.
(B) Set up, without solving, the binomial probability P(x is at most 4) using probability notation.
(C) How would you find the normal approximation to the binomial probability P(x = 4) in part A? Please show how you would calculate µ and σ in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations.

User Icon for latoya latoya answered
12 years ago

=0.2311

(B) p(x is at most 4)=P (x,=4)=p(0)+p(1)+p(2)+p(3)+p(4)
=c(12,0)(0.30)^0(1-0.30)^2(12-0)+c(12,1)(0.30)^1(1-0.30)^(12-1)
+c(12,2)(0.30)^2(1-0.30)^(12-2)+c(12,3)(0.30)^3(1-0.30)^(12-3)+c(12,4)(0.30)^4(1-0.30)^(12-4)

(C) u=np=12*0.3=3.6
12*0.3*0.7=2.52
(2.52)=1.5875
to find p(4), we applies the continuity correction factor and find p(3.5<x<4.5). this because using the normal distribution p(x=4) will be 0.
z=(3.4-3.6)/1.5875=-0.0630and z=(4.5-3.6)/1.5875=0.5669
P(4)= P(3.5<x<4.5)=P(-0.063<z<0.5669)=0.2397

User Icon for Explain Bot Explain Bot answered
11 months ago

(A) To find the binomial probability P(x = 4), where n = 12 and p = 0.30, we can use the formula for binomial probability:

P(x) = (nCx) * p^x * (1-p)^(n-x)

where nCx represents "n choose x", and it can be calculated using the formula:

nCx = n! / (x! * (n-x)!)

In this case, n = 12, x = 4, and p = 0.30. Let's calculate P(x = 4):

P(4) = (12C4) * (0.30^4) * (0.70^8)

Calculating the values:

12C4 = 12! / (4! * (12-4)!) = 495

P(4) = 495 * (0.30^4) * (0.70^8) = 0.152

So, P(x = 4) is approximately 0.152.

(B) To set up the binomial probability P(x is at most 4) using probability notation, we can write:

P(x ≤ 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4)

This represents the sum of the probabilities of all possible outcomes from x = 0 to x = 4.

(C) To find the normal approximation to the binomial probability P(x = 4) in part A, we can use the formula for the mean (µ) and standard deviation (σ) of a binomial distribution, which are given by:

µ = n * p
σ = √(n * p * (1-p))

In this case, n = 12 and p = 0.30. Let's calculate µ and σ:

µ = 12 * 0.30 = 3.6
σ = √(12 * 0.30 * (1-0.30)) ≈ 1.799

Now, we can use these values to find the normal approximation for P(x = 4) using the formula for the normal distribution:

P(x = 4) ≈ P(3.5 < x < 4.5)

So, the final formula for the normal approximation to the binomial probability P(x = 4) is:

P(3.5 < x < 4.5) = Φ((4.5 - µ) / σ) - Φ((3.5 - µ) / σ)

where Φ represents the cumulative distribution function of the standard normal distribution.