Chapter 2.6, Problem 1E

To determine

Identify whether the events $E_1$ and $E_2$ are independent or not.

Answer to Problem 1E

The events $E_1$ and $E_2$ are not independent.

Explanation of Solution

Independent events:

Let A and B be two events are said to be independent if, if occurrence of event A does not influence occurrence of events B then events are referred as independent. The expression is, $P\left(B|A\right)=P\left(B\right)$

The event $E_1$ denotes the first diode selected has efficiency below 0.28, and $E_2$ denotes the second diode selected has efficiency above 0.35.

A sample of 10 laser diodes in which two have efficiency below 0.28, six have efficiency between 0.28 and 0.35, and two have efficiency above 0.35.

Two diodes are selected at random and without replacement.

The probability that the second diode selected has efficiency above 0.35 is,

$P\left(E_2\right)=\frac{2}{10}$

The probability that the second diode selected has efficiency above 0.35 given that the first diode selected has efficiency below 0.28 is,

$P\left(E_2|E_1\right)=\frac{2}{9}$

It is clear that, $P\left(E_2|E_1\right)\ne P\left(E_2\right)$ and events are not independent.

Hence, the events $E_1$ and $E_2$ are not independent.