Binomial distribution examples and solutions

Answer

Binomial Distribution problems worksheet

Example 1

It is expected that 10% of production from a continous process will be defective.

Find the probability that in a sample of 10 units chosen at random exactly 2 will be defective and atleast 2 will be defective.

Solution

Exactly 2 will be defective;

$P(X=2) =\left(\begin{array}{c}10\\ 2\end{array}\right){(0.1)}^2{(0.9)}^{10-2}$ 

$=\left(\begin{array}{c}10\\ 2\end{array}\right){(0.1)}^2{(0.9)}^8$

$$\begin{gathered} =45\times 0.01\times 0.43047 \\ =0.1937 \end{gathered}$$

Atleast 2 will be defective

$P(x\ge 2)= P(x=2)+P(x=3)+........P(x=10)$

$=1-P(x<2)$

$P(X\ge 2) =1-\{ \left(\begin{array}{c}10\\ 0\end{array}\right){(0.1)}^0{(0.9)}^{10} + \left(\begin{array}{c}10\\ 1\end{array}\right){(0.1)}^1{(0.9)}^9 \}$

$=1-(0.3487+0.3874)$

=0.2640

Example 2

Probability that a computer drawn from a batch of computer its defective is 0.1. If a sample of 6 computer is taken. Find the probability that it will contain;

(i) No defective computer

$P(X=0) =\left(\begin{array}{c}6\\ 0\end{array}\right){(0.1)}^0{(0.9)}^{10}$

$=1\times 1\times 0.34867844$

=0.3487

(ii) 5 or 6 are defective Computer

$P(X<5) =\left(\begin{array}{c}6\\ 5\end{array}\right){(0.1)}^5{(0.9)}^1 + \left(\begin{array}{c}6\\ 6\end{array}\right){(0.1)}^6{(0.9)}^0$

=0.000054+0.000001

=0.000055

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