- b( 3, 8, 0.6 )
- b( 5, 8, 0.6 )
- P( 3 $\le$ X $\le$ 5 ) when n = 8 and p = 0.6
The aim of this question is to use the binomial random variable and its probability mass function to find probability values.
The binomial probability mass function is mathematically defined as:
\[ P( \ X \ = \ x \ ) \ = \ b( \ x, \ n, \ p \ ) \ = \ \left ( \begin{array}{c} n \\ x \end{array} \right ) \ p^x \ ( \ 1 \ – \ p \ )^{ n – x } \]
Expert Answer
Part (a) – b( 3, 8, 0.6 )
\[ b( \ 3, \ 8, \ 0.6 \ ) \ = \ \left ( \begin{array}{c} 8 \\ 3 \end{array} \right ) \ (0.6)^3 \ ( \ 1 \ – \ 0.6 \ )^{ 8 – 3 } \]
\[ b( \ 3, \ 8, \ 0.6 \ ) \ = \ \dfrac{ 8! }{ 3! \ (8 – 3)! } \ (0.6)^3 \ ( \ 0.4 \ )^5 \]
\[ b( \ 3, \ 8, \ 0.6 \ ) \ = \ \dfrac{ 8! }{ 3! \ 5! } \ (0.6)^3 \ (0.4)^5 \]
\[ b( \ 5, \ 8, \ 0.6 \ ) \ = \ (56) \ (0.6)^3 \ (0.4)^5 \]
\[ b( \ 3, \ 8, \ 0.6 \ ) \ = \ 0.1238 \]
– b( 5, 8, 0.6 )
\[ b( \ 5, \ 8, \ 0.6 \ ) \ = \ \left ( \begin{array}{c} 8 \\ 5 \end{array} \right ) \ (0.6)^5 \ ( \ 1 \ – \ 0.6 \ )^{ 8 – 5 } \]
\[ b( \ 5, \ 8, \ 0.6 \ ) \ = \ \dfrac{ 8! }{ 5! \ (8 – 5)! } \ (0.6)^5 \ ( \ 0.4 \ )^3 \]
\[ b( \ 5, \ 8, \ 0.6 \ ) \ = \ \dfrac{ 8! }{ 5! \ 3! } \ (0.6)^3 \ (0.4)^5 \]
\[ b( \ 5, \ 8, \ 0.6 \ ) \ = \ (56) \ (0.6)^5 \ (0.4)^3 \]
\[ b( \ 5, \ 8, \ 0.6 \ ) \ = \ 0.2787 \]
– P( 3 $\le$ X $\le$ 5 ) when n = 8 and p = 0.6
Using same approach as part (a) and (b):
\[ P( \ X \ = \ 4 \ ) \ = \ b( \ 4, \ 8, \ 0.6 \ ) \ = \ 0.2322 \]
Since:
\[ P( \ 3 \le X \le 5 \ ) \ = \ P( \ X \ = \ 3 \ ) \ + \ P( \ X \ = \ 4 \ ) \ + \ P( \ X \ = \ 5 \ ) \]
\[ P( \ 3 \le X \le 5 \ ) \ = \ 0.1238 \ + \ 0.2322 \ + \ 0.2787 \]
Numerical Result
b( 3, 8, 0.6 ) = 0.1238
b( 5, 8, 0.6 ) = 0.2787
P( 3 $\le$ X $\le$ 5 ) = 0.6347
Example
Find the probability P( 1 $\le$ X ) where X is a random variable with n = 12 and p = 0.1
Using same approach as part (a) and (b):
\[ P( \ X \ = \ 0 \ ) \ = \ b( \ 0, \ 12, \ 0.1 \ ) \ = \ 0.2824 \]
Since:
\[ P( \ 1 \le X \ ) \ = \ 1 \ – \ P( \ X \le 1 \ ) \ = \ 1 \ – \ P( \ X \ = \ 0 \ ) \]
\[ P( \ 1 \le X \ ) \ = \ 1 \ – \ 0.2824 \ = \ 0.7176 \]