11Bmorebinomial


 * Binomial Probabilities for a Given Range **

You should be familiar with the following phrases:

"X is more than 3" means X > 3 "X is greater than 3" means X > 3

"X is at least 3" means X __>__ 3 "X is not less than 3" mean X __>__ 3

"X is no more than 3" means X __<__ 3 "X is up to and including 3" means X __<__ 3

"X is less than 3" means X < 3


 * Example 1 **

Given the following probability distribution:
 * a) Find the probability that X is more than 3 **

math \\ . \qquad Pr(X > 3) = Pr(X=4) + Pr(X = 5) \qquad .\\. \\ . \qquad \qquad \qquad \; = 0.0112 + 0.0048 \\. \\ . \qquad \qquad \qquad \; = 0.0160 math


 * b) Find the probability that X is no more than 4 **

math \\ . \qquad Pr(X \leqslant 4) = Pr(X=0) + Pr(X=1) + Pr(X=2) + Pr(X=3) + Pr(X=4) \qquad .\\. \\ . \qquad \qquad \qquad \; = 0.2311 + 0.3147 + 0.3321 + 0.1061 + 0.0112 \\. \\ . \qquad \qquad \qquad \; = 0.9952 math


 * NOTE: There is an __easier__ way to do part b**

math \\ . \qquad Pr(X \leqslant 4) = 1 - Pr(X = 5) \qquad. \\ . \\ . \qquad \qquad \qquad \; = 1 - 0.0048 \\. \\ . \qquad \qquad \qquad \; = 0.9952 math

CAS calculator From the main screen, go to the ** Interactive ** menu and select ** Distribution **.

Recall that selecting **Binomial PDF** will calculate Pr(X = x)

In the same way, selecting ** Binomial CDF ** will calculate Pr(a __<__ X __<__ b) {CDF stands for Cumulative Distribution Function}

Enter {lower limit, upper limit, number trials, p(x) }

Note: Some older versions of the classpad will only calculate Pr(X __<__ a)

Given that X ~ Bi(6, 0.3)
 * Example 1 **


 * a) Pr (X < 3)**

math \\ . \qquad Pr (X < 3) = Pr (X \leqslant 2) \\. \\ . \qquad \qquad \textbf{Enter: } binomialcdf \;\; lwr = 0,\; upr = 2, n = 6, p = 0.3 \qquad. \\ . \\ . \qquad = 0.744 math

math \\ \textbf{b) Pr (X } \geqslant \textbf{ 3)} \\ . \\ . \\ . \qquad \qquad \textbf{Enter: } binomialcdf \;\; lwr = 3, upr = 6, n = 6, p = 0.3 \qquad . \\ . \\ . \qquad = 0.256 math


 * c) Pr (2 < X < 4)**

math \\ . \qquad = Pr(X = 3) \\. \\ . \qquad \qquad \textbf{enter: } binomialpdf \;\; x = 3, n = 6, p = 0.3 \qquad. \\ . \\ . \qquad = 0.185 math

math \\ \textbf{d) Pr ( 2 } \leqslant \textbf{ X } \leqslant \textbf{ 4)} \qquad . \\ . \\ . \\ . \qquad \qquad \textbf{Enter: } binomialcdf \;\; lwr = 2, upr = 4, n = 6, p = 0.3 \qquad . \\ . \\ . \qquad = 0.569 math

Using Properties of Distributions

We already know that for any distribution (including the Binomial Distribution), it is true that math . \qquad Pr(X \geqslant a) = 1 - Pr(X < a) \qquad. math

This means that (for a discrete distribution) math . \qquad Pr(X \geqslant a) = 1 - Pr \big( X \leqslant (a-1) \big) \qquad. math

math . \qquad Pr(X \geqslant 5) = 1 - Pr(X \leqslant 4) \qquad. math
 * Eg **

Consideration of the diagram shows that (if a < b) math \\ . \qquad Pr(a \leqslant X \leqslant b) = Pr(X \leqslant b) - Pr(X < a) \\ \textit{so:} \\ . \qquad Pr(a \leqslant X \leqslant b) = Pr(X \leqslant b) - Pr \big( X \leqslant (a - 1) \big) \qquad. math
 * Also **

math . \qquad Pr(5 \leqslant X \leqslant 8) = Pr(X \leqslant 8) - Pr(X \leqslant 4) \qquad. math
 * Eg **

math . \qquad Pr \big( \, (X \leqslant a) \cap (X \leqslant b) \, \big) = Pr(X \leqslant a) \qquad. math
 * Also **

math . \qquad Pr \big( (X \leqslant 4) \cap (X \leqslant 8) \, \big) = Pr(X \leqslant 4) \qquad. math
 * Eg **

{Calculator Free Question}
 * Example 2 **

In a certain binomial distribution math . \qquad Pr(X \leqslant 4) = 0.1 \text{ and } Pr(X \leqslant 10) = 0.8 \text{ then: } \qquad. math


 * a) Find Pr(X __>__ 5) **

math \\ . \qquad Pr(X \geqslant 5) \\. \\ . \qquad = 1 - Pr(X \leqslant 4) \qquad. \\ . \\ . \qquad = 1 - 0.1 \\. \\ . \qquad = 0.9 math


 * b) Find Pr(5 __<__ X __<__ 10) **

math \\ . \qquad Pr(5 \leqslant X \leqslant 10) \\. \\ . \qquad =Pr(X \leqslant 10) - Pr(X < 5) \\. \\ . \qquad = Pr(X \leqslant 10) - Pr(X \leqslant 4) \qquad. \\ . \\ . \qquad = 0.8 - 0.1 \\. \\ . \qquad = 0.7 math


 * d) Find Pr(X __<__ 4 | X __<__ 10) **

math \\ . \qquad Pr(X \leqslant 4 \; | \; X \leqslant 10) \\. \\ . \qquad = \dfrac{Pr \big( (X \leqslant 4) \cap (X \leqslant 10) \big) }{Pr(X \leqslant 10)} \qquad. \\ . \\ . \qquad = \dfrac{Pr(X \leqslant 4)}{Pr(X \leqslant 10)} \\. \\ . \qquad = \dfrac{0.1}{0.8} \\. \\ . \qquad = \dfrac{1}{8} math

.