###### Decision Rules in Hypothesis Tests

The decision rule refers to the procedure followed by analysts and researchers when... **Read More**

Two or more events are independent if the occurrence of one event does not influence the occurrence of the other event(s). Let us put this in annotations:

$$ \text P ( \text A | \text B) = \text P (\text A) $$

or

$$ \text P (\text B | \text A) = \text P (\text B) $$

The events are said to be dependent if this condition is not satisfied.

Suppose you rolled a die and flipped a coin. The probability of getting any number face on the die does not influence the probability of getting a head or a tail on the coin. The two events are independent.

Suppose you have a bag containing 8 green balls and 5 blue balls. If you draw a blue ball without replacing it, the probability of drawing another blue ball on your second attempt is greatly changed because you drew a blue ball the first time. Such events are said to be dependent.

If two events are independent:

$$ \text P (\text {AB}) = \text P (\text A) × \text P (\text B) $$

QuestionMike Jamerson tosses a coin twice. What is the probability of getting a head on the first toss and a head on the second toss?

A. \(\frac{1}{2}\).

B. \(\frac{1}{4}\).

C. \(\frac{2}{3}\).

SolutionThe correct answer is

B.The coin has no memory. As such, the first outcome has no influence on the second outcome. Therefore:

$$ \begin {align*}

\text P (\text {Head and Head}) & = \text P (\text H) × \text P (\text H) \\

& = \cfrac {1} {2} × \cfrac {1} {2} \\

& = \cfrac {1} {4} \\

\end {align*} $$