A box contains 30 raffle tickets numbered 1 through 30. If one ticket is selected at rando… — ACT

A box contains $30$ raffle tickets numbered $1$ through $30$ . If one ticket is selected at random from the box, what is the probability that the number on the selected ticket is a multiple of $4$ or a multiple of $6$ ?

A
$\frac{1}{15}$
B
$\frac{1}{5}$
$\frac{1}{3}$ Answer
D
$\frac{2}{5}$
E
$\frac{1}{2}$

Answer

The probability that the number on the selected ticket is a multiple of

4

or a multiple of

6

\frac{1}{3}

The correct answer is determined by applying the addition rule for non-mutually exclusive events:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

. Out of

30

tickets,

7

are multiples of

4

and

5

are multiples of

6

. The numbers

12

and

24

are multiples of both

4

and

6

, meaning they belong to the intersection. Calculating the total number of unique favorable tickets gives

7 + 5 - 2 = 10

. Dividing

10

favorable outcomes by the

30

total tickets yields a probability of

\frac{10}{30} = \frac{1}{3}

Step-by-Step Solution

Identify the total number of possible outcomes.

There are

30

total tickets in the box, so the sample space size is

30

Each ticket numbered 1 through 30 is equally likely to be selected.

Count the outcomes for each individual event.

Multiples of

4

\{4, 8, 12, 16, 20, 24, 28\} \rightarrow 7

outcomes.
Multiples of

6

\{6, 12, 18, 24, 30\} \rightarrow 5

outcomes.

Listing or dividing

30

4

and

6

determines how many tickets satisfy each separate condition.

Identify and count the overlapping outcomes (multiples of both

4

and

6

Multiples of both

4

and

6

are multiples of

\text{LCM}(4, 6) = 12

, which are

\{12, 24\} \rightarrow 2

outcomes.

These numbers are contained in both individual lists and will be double-counted if not accounted for.

Apply the Principle of Inclusion-Exclusion to find the total number of favorable outcomes.

\text{Favorable outcomes} = 7 + 5 - 2 = 10.

Subtracting the intersection prevents double-counting the tickets that satisfy both conditions.

Calculate the probability.

P(\text{multiple of } 4 \text{ or } 6) = \frac{10}{30} = \frac{1}{3}.

Probability is the ratio of favorable outcomes to total possible outcomes.

Key Concept

Probability of Non-Mutually Exclusive Events (Inclusion-Exclusion Principle)

Estimated Time:1m 0s

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