Question

A probability that a bulb produced by a factory will fuse after $150$ days of use is $0.05$. What is the probability that out of $5$ such bulbs none will fuse after $150$ days of use

• A. $1-{\left(\frac{19}{20}\right)}^5$
• B. ${\left(\frac{19}{20}\right)}^5$
• C. ${\left(\frac{3}{4}\right)}^5$
• D. $90{\left(\frac{1}{4}\right)}^5$

Answer 1

The correct option is B

${\left(\frac{19}{20}\right)}^5$

Explanation for the correct option:

Step 1: Find the probability that the bulb will not fuse:

Subtract the probability of bulb will fuse to unity.

Given: The probability of bulb will fuse

$\begin{array}{rcl}& =& 0.05\\ & =& \frac{1}{20}\end{array}$

So, the probability that the bulb will not fuse

$$\begin{gathered} =1-\frac{1}{20} \\ =\frac{19}{20} \end{gathered}$$

Step 2: Use binomial distribution to find the required probability:

Find the binomial coefficient for $n=5,r=5$

$$\begin{gathered} p={}^{5}C_5{\left(\frac{19}{20}\right)}^5{\left(\frac{1}{20}\right)}^0 \\ =1{\left(\frac{19}{20}\right)}^5\left(1\right) \\ ={\left(\frac{19}{20}\right)}^5 \end{gathered}$$

Hence, option (B) is the correct answer.