In a class of 25 students, 15 of them have a cat,
16 of them have a dog and 3 of them have neither.
Find the probability that a student chosen at random
has a cat and a dog.

P(A or B) = P(A) + P(B) ⇒ mutually exclusive (events cannot happen at the same time)
P(A or B) = P(A) + P(B) - P(A and B) ⇒ non-mutually exclusive (if they have at least one outcome in common)

The rule of probability is P(A) = [tex]\frac{n(A)}{n}[/tex]

∵ A class has 25 students

∵ 15 of them have a cat

∴ n(cat) = 15

- By using the rule of probability

∴ P(cat) = [tex]\frac{15}{25}[/tex]

∵ 16 of them have a dog

∴ n(dog) = 16

- By using the rule of probability

∴ P(dog) = [tex]\frac{16}{25}[/tex]

∵ 3 of them have neither

- Subtract from the total of the class to find the number of

students that have a cat or a dog

∴ n(cat or dog) = 25 - 3

∴ n(cat or dog) = 22

- By using the rule of probability

∴ P(cat or dog) = [tex]\frac{22}{25}[/tex]

To find the probability that a student has a cat and a dog use the non-mutually rule of addition above

∵ P(cat or dog) = P(cat) + P(dog) - P(cat and dog)

∴ [tex]\frac{22}{25}[/tex] = [tex]\frac{15}{25}[/tex] + [tex]\frac{16}{25}[/tex] - P(cat and dog)

- Add the like terms in the right hand side

∴ [tex]\frac{22}{25}[/tex] = [tex]\frac{31}{25}[/tex] - P(cat and dog)

- Subtract [tex]\frac{31}{25}[/tex] from both sides

∴ [tex]-\frac{9}{25}[/tex] = - P(cat and dog)

- Multiply both sides by -1

∴ [tex]\frac{9}{25}[/tex] = P(cat and dog)

- Switch the two sides

∴ P(cat and dog) = [tex]\frac{9}{25}[/tex]

The probability that a student chosen at random has a cat and a dog is [tex]\frac{9}{25}[/tex]

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