Exercise 2 Page 677

We know that a group of the students includes one boy and three girls. The teacher randomly selects one of the students to be the speaker and different student to be the recorder. We want to determine whether randomly selecting a girl first and randomly selecting a boy second 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. $P (A and B) = P (A) \cdot P (B)$

We will begin by recalling the outcomes in the sample space. Let M represent the boy and $G_{1},$ $G_{2},$ and $G_{3}$ represent the three girls.

Number of Girls	Outcomes
$1$	$G_{1}$ M M $G_{1}$
$1$	$G_{2}$ M M $G_{2}$
$1$	$G_{3}$ M M $G_{3}$
$2$	$G_{1} G_{2}$ $G_{2} G_{1}$
$2$	$G_{1} G_{3}$ $G_{3} G_{1}$
$2$	$G_{2} G_{3}$ $G_{3} G_{2}$

In our case, we have two events given.

$A$ — randomly selecting a girl first.
$B$ — randomly selecting a boy second.

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.

P = \frac{Favorable Outcomes}{Possible Outcomes}

From the table above we can see that there are

9

outcomes in which the teacher picks female student first and the number of possible outcomes is

12 .

Therefore, we are able to obtain

P (A) .

P = \frac{Favorable Outcomes}{Possible Outcomes}

SubstituteValues

Substitute values

P (A) = \frac{9}{1 2}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

P (A) = \frac{3}{4}

Now let's find the probability of event

B .

This time the number of favorable outcomes is

3 .

The number of possible outcomes again is equal to

12 .

P = \frac{Favorable Outcomes}{Possible Outcomes}

SubstituteValues

Substitute values

P (B) = \frac{3}{1 2}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

P (B) = \frac{1}{4}

Lastly, we need to calculate the probability of the event

A and B,

which represents the situation in which the teacher randomly selects a girl first and a boy second. Using the table, we know the number of favorable outcomes is

3

and the possible outcomes is

12 .

P = \frac{Favorable Outcomes}{Possible Outcomes}

SubstituteValues

Substitute values

P (A and B) = \frac{3}{1 2}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

P (A and B) = \frac{1}{4}

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.

P (A and B) = P (A) \cdot P (B)

Let's substitute

\frac{3}{4}

for

P (A)

and

\frac{1}{4}

for

P (B)

and compare the product with previously found

P (A and B) .

P (A and B) = ? P (A) \cdot P (B)

SubstituteII

$P (A) = \frac{3}{4}$ , $P (B) = \frac{1}{4}$

P (A and B) = ? \frac{3}{4} \cdot \frac{1}{4}

Substitute

$P (A and B) = \frac{1}{4}$

\frac{1}{4} = ? \frac{3}{4} \cdot \frac{1}{4}

▼

Evaluate

MultFrac

Multiply fractions

\frac{1}{4} = ? \frac{3 \cdot 1}{4 \cdot 4}

Multiply

\frac{1}{4} = ? \frac{3}{1 6}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 4}{b \cdot 4}$

\frac{4}{1 6} = ? \frac{3}{1 6} \times

Since we have not obtained the identity

P (A and B) = P (A) \cdot P (B),

the teacher randomly selecting a girl first and randomly selecting a boy second are not independent events.