We are told that there are $40$ students, $9$ camp counselors, and $5$ teachers at Camp Kern. There are three activities that a person can do: hiking, horseback riding, and a canoe trip. Each person is assigned to only one of these activities. Let's make a two-way table to organize the given information.

	Hiking	Horseback Riding	Canoe Trip	Total
Student	$9$	$17$		$40$
Camp Counselor	$2$		$3$	$9$
Teacher		$2$	$2$	$5$

Since we know the total number of students, camp counselors, and teachers, we can calculate the missing data.

	Hiking	Horseback Riding	Canoe Trip	Total
Student	$9$	$17$	$14$	$40$
Camp Counselor	$2$	$4$	$3$	$9$
Teacher	$1$	$2$	$2$	$5$

We will find the probability of a randomly selected person being a student on the canoe trip

A

or a camp counselor horseback riding

B .

The events

A

and

B

are mutually exclusive. Therefore, the probability of selecting a student on the canoe trip

A

or a camp counselor on a horse

B

is the sum of their individual probabilities.

P (A or B) = P (A) + P (B)

Let's first calculate the probability of

A .

This probability can be found by the ratio of the number of students on the canoe trip to the total number of people at the camp.

P (A) = \frac{students on the canoe trip}{total number of people}

Looking at the table, we will substitute the values into this ratio and calculate it.

P (A) = \frac{Students on the canoe trip}{Total number of people}

SubstituteValues

Substitute values

P (A) = \frac{1 4}{4 0 + 9 + 5}

AddTerms

Add terms

P (A) = \frac{1 4}{5 4}

Let's now calculate the probability of a randomly selected person being a camp counselor riding a horse

B

by the same method used to find

P (A) .

P (B) = \frac{Camp counselors on a horse}{Total number of people}

Using the table, we will substitute the values into the ratio and calculate

P (B) .

P (B) = \frac{Camp counselors on a horse}{Total number of people}

SubstituteValues

Substitute values

P (B) = \frac{4}{4 0 + 9 + 5}

AddTerms

Add terms

P (B) = \frac{4}{5 4}

Finally, we will substitute the ratios

P (A) = \frac{1 4}{5 4}

and

P (B) = \frac{4}{5 4}

into the equation and solve it for

P (A or B) .

P (A or B) = P (A) + P (B)

SubstituteII

$P (A) = \frac{1 4}{5 4}$ , $P (B) = \frac{4}{5 4}$

P (A or B) = \frac{1 4}{5 4} + \frac{4}{5 4}

AddFrac

Add fractions

P (A or B) = \frac{1 8}{5 4}

DivFracByFracSameDenom

$\frac{a / 1 8}{b / 1 8} = \frac{a}{b}$

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

The probability that a randomly selected person will be a student on a canoe trip or a camp counselor riding a horse is

\frac{1}{3} .

Exercise