Question

Let E be the event that a corn crop has an infestation of earworms, and let B be the event that a corn crop has an investigation of corn borers.

Answer 1

We will see how to determine independent probabilities of two events with the help of set notations.

We will define two events that can occur independently as follows:

`\begin{gathered} \text{Event ( E ) = Corn crop infected by earth worms} \\ \text{Event ( B ) = Corn crop infected by corn borers} \end{gathered}`

The probability of occurrence of each event is defined by the prefix " p " preceeding each event notation as follows:

`\begin{gathered} p\text{ ( E ) = 0.18} \\ p\text{ ( B ) = 0.11} \\ \text{p ( E \& B ) = 0.06} \end{gathered}`

We can express the two events ( E and B ) as two sets. We will Venn diagram to express the events as follows:

We have expressed the two events E and B as circles which are intersecting. The region of intersection is the common between both events ( E & B ).

The required probability is defined as an instance when the corn crop is subjected to either earth worm or corn borers or both!

If we consider Venn diagram above we can see the required region constitutes of region defined by event ( E ) and event ( B ). We can sum up the regions defined by each event!

However, if we consider region E and B as stand alone we see that the common region is added twice in the algebraic sum of induvidual region E and B. We will discount the intersection region once to prevent over-counting. Therefore,

`p\text{ ( E U B ) = p ( E ) + p ( B ) - p ( E \& B )}`

The above rule is the also denoted as the rule of independent events ( probabilities ).

We will use the above rule to determine the required probability p ( E U B ) as follows:

`\begin{gathered} p\text{ ( E U B ) = 0.18 + 0.11 - 0.06} \\ p\text{ ( E U B ) = 0.23} \end{gathered}`

Therefore, the required probability is:

`0.23\ldots\text{ Option D}`