Question

Bulbs are packed in cartons each containing \[40\] bulbs. Seven hundred cartons were examined for defective bulbs and the results are in the following table:

Number of defective bulbs in a cartoon	Frequency
$0$	$400$
$1$	$180$
$2$	$48$
$3$	$41$
$4$	$18$
$5$	$8$
$6$	$3$
More than $6$	$2$

One carton was selected at random. What is the probability that it has
1. No defective bulb?
2. Defective bulbs from $2$ to $6$?
3. Defective bulbs less than $4$?

Answer 1

Hint: Here, in the given question, we are given that bulbs are packed in cartons each containing $40$ bulbs and seven hundred cartoons were examined for defective bulbs and the results are given in the table and we need to find the probability of different cases. Probability is a measure of the likelihood of an event to occur. The probability formula is defined as the probability of an event to happen is equal to the ratio of the number of favorable outcomes and the total number of outcomes. We find the probability of each case using the formula of probability.

Formula used:
 For any event $A$, $P\left( A \right) = \dfrac{{Number{\text{ }}of{\text{ }}favourable{\text{ }}outcome}}{{Total{\text{ }}number{\text{ }}of{\text{ }}favourable{\text{ }}outcomes}}$

Complete answer:
1. No defective bulb?
Favourable outcomes = $400$
$P$ (Carton has no defective bulb) = $\dfrac{{400}}{{700}} = \dfrac{4}{7}$

2. Defective bulbs from $2$ to $6$?
Defective bulbs from $2$ to $6$ = $2$ or $3$ or $4$ or $5$ or $6$ defective bulbs
Favourable outcomes = $48 + 41 + 18 + 8 + 3 = 118$
$P$ (Defective bulb from $2$ to $6$) = \[\dfrac{{118}}{{700}} = \dfrac{{59}}{{350}}\]

3. Defective bulbs less than $4$?
Defective bulbs less than $4$ = Defective bulbs equal to $0$ or $1$ or $2$ or $3$.
Favourable outcomes = $400 + 180 + 48 + 41 = 669$
$P$ (Defective bulb less than $4$) = \[\dfrac{{669}}{{700}}\]

Note: Here, it is mentioned in the question that the selection is random, it means when you randomly select an object out of $n$ objects, each of the $n$ objects has the same probability of being chosen. You didn’t say that $n$ is the total number, but if that is what you mean, then it’s picked out of the total number of objects. If $n$ is less than the total, you get a different result. The most important part in these types of questions is to find out the number of ways to select an object.