Question

If 20% of the bolts produced by a machine are defective, the probability that out of 4 bolts chosen at random, less than 2 bolts will be defective is

Answer 1

Answer:

Probability of less than 2 defective bolt out of 4 randomly choose is $\frac{297}{625}$

Step-by-step explanation:

Given: 20% of bolts are defective

To find: Probability of less than 2 bolts defective.

Probability of getting a defective bolt = $\frac{40}{100}\,=\,\frac{2}{5}$

Probability of getting a good bolt = $1-\frac{2}{5}\,=\,\frac{3}{5}$

Let X represent the no of defective bolts.

We have to find probability of less than 2 defective bolts.

⇒ P [ X < 2 ] = P [ X = 0 ] + P [ X = 1 ]

So, P [ X = 0 ] = P [ 0 defective bolt ] = $\frac{3}{5}\times\frac{3}{5}\times\frac{3}{5}\times\frac{3}{5}$

                      = $\frac{81}{625}$

P [ X = 1 ] = P [ 1 defective bolt]  = $4\times\frac{2}{5}\times\frac{3}{5}\times\frac{3}{5}\times\frac{3}{5}$

               = $\frac{216}{625}$

⇒ P [ X < 2] = $\frac{81}{625}+\frac{216}{625}$

                   = $\frac{297}{625}$

Therefore, Probability of less than 2 defective bolt out of 4 randomly choose is $\frac{297}{625}$

Answer 2

The probability of getting less than 2 defective bolt out of 4 randomly chosen bolts is $\frac{26}{125}$.

Multiplication  Theorem

If A and B are two independent events, then the probability that both will occur is equal to the product of their individual probabilities.

Given: $20$% of bolts  produced by machine are defective.

Explanation

Probability of getting a defective bolt is $20$%,  $\frac{40}{100}=\frac{2}{5}$.

Then, the probability of getting a good bolt is $80$% , $1-\frac{2}{5} =\frac{3}{5}$

Let ' $X$' be the no of defective bolts.

To find probability of less than 2 defective bolts.

P [ X < 2 ] = P [ X = 0 ] + P [ X = 1 ]

P [ No defective bolt ] , P [ X = 0 ] , $\frac{3}{5}$×  $\frac{3}{5}$×$\frac{3}{5}$×$\frac{3}{5}$ = $\frac{81}{625}$

P [ 1 defective bolt], P [ X = 1 ] ,  $\frac{2}{5}$×$\frac{3}{5}$×$\frac{3}{5}$×$\frac{3}{5}$ = $\frac{54}{625}$

Use equation to solve the total probability,

P [ X < 2]  = P [ X = 0 ] + P [ X = 1 ]

 = $\frac{81}{625}$ + $\frac{54}{625}$

P [ X < 2]  is $\frac{135}{625}$ = $\frac{26}{125}$

Hence, the probability of less than 2 defective bolt out of 4 randomly chosen bolts  is  $\frac{26}{125}$.

To know more about multiplication theorem, here

https://brainly.in/question/4042362?msp_poc_exp=2

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