Exercise 1 Page 677

We know that a student taking a quiz randomly guesses the answers to four true-false questions. We want to determine whether guessing Question $1$ incorrectly and guessing Question 2 correctly are independent events. Let's recall the Probability of Independent Events formula.

Probability of Independent Events

Two events A and B are independent events if and only if the probability that both events occur is the product of the probabilities of the events. $P(A\text{ and }B)=P(A)\cdot P(B)$

Let's write out the possible outcomes in the sample space. Let $\text{C}$ represent a correct answer and $\text{I}$ represent an incorrect answer.

Number Correct	Outcomes
$0$	$\text{IIII}$
$1$	$\text{C}\text{III }$ $\text{I}\text{C}\text{II }$ $\text{II}\text{C}\text{I }$ $\text{III}\text{C}$
$2$	$\text{II}\text{CC }$ $\text{I}\text{C}\text{I}\text{C }$ $\text{I}\text{CC}\text{I }$ $\text{C}\text{II}\text{C }$ $\text{C}\text{I}\text{C}\text{I }$ $\text{CC}\text{II}$
$3$	$\text{I}\text{CCC }$ $\text{C}\text{I}\text{CC }$ $\text{CC}\text{I}\text{C }$ $\text{CCC}\text{I}$
$4$	$\text{CCCC}$

In our case we have two events given.

• $A:$ guessing Question $1$ incorrectly.
• $B:$ guessing Question $2$ correctly.

In order to find the probability of event $A,$ we will use the theoretical probability. We need to find the ratio of the number of favorable outcomes to the number of possible outcomes. From the table above we can see that there are $8$ outcomes in which the student guesses Question $1$ incorrectly, and the number of possible outcomes is $16.$ Therefore, we are able to calculate $P(A).$ Now, let's find the probability of event $B.$ From the table we can see that both the number of favorable and possible outcomes remain the same as for event $A.$ Lastly, we can calculate the probability of event $A\text{ and }B,$ which represents the situation in which the student is incorrect on Question $1$ and correct on Question $2.$ Using the table, we know the number of favorable outcomes is $4$ and the number of possible outcomes is $16.$ In order to determine whether two events $A$ and $B$ are independent, we need to multiply $P(A)$ and $P(B)$ and check whether the identity below holds. If and only if it does are the events independent. Let's substitute $\frac{1}{2}$ for $P(A)$ and $\frac{1}{2}$ for $P(B)$ and compare their product with previously found $P(A\text{ and }B).$ Since the identity $P(A\text{ and }B)=P(A)\cdot P(B)$ holds true, guessing Question $1$ incorrectly and guessing Question 2 correctly are independent events.