Geometry Question

Question

each of the numbers from 1 to 30 is written on a card and placed in a bag. if one card is drawn at random, what is the probability that the number is a multiple of 2 or a multiple of 3?

Answer 1

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

Either a multiple of 2 or a multiple of 3:

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30

Not a multiple of either 2 or 3: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29

P = 20/30 = 2/3 = 0.67

PLEASE DOUBLE-CHECK YOUR ANSWER (Everyone makes typos).

Answer 2

(Easy to understand solution)

When we calculate the probability of something happening, we find the number of ways it can happen as a fraction of the total number of outcomes.

We make this formal in mathematics by representing the possible outcomes as a set, and particular events as subsets of this set.

For example, the flip of a coin could be represented by the set of two outcomes:

S = { H, T }

and the particular event of flipping heads by the set of just the one outcome:

E = { H }.

Using this formalism allows us to express the probability of the event E, i.e. flipping heads, as follows:

P(E) = | E | / | S | = 1 / 2 = 0.5

(where |A| denotes the number of elements in the set A)

Your question can be answered using the same technique.

S = { 1, 2, 3, ... , 29, 30 } and

E = { multiples of 2 and 3 in S } = { 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30 }

and so

P(drawing a multiple of 2 or 3) = P(E) = | E | / | S | = 20 / 30 = 2 / 3 or about 0.667

Hope that helps!

(More advanced solution and bonus visual)

Let A represent drawing a multiple of 2, and B drawing a multiple of 3.

P(A or B)

= P(A) + P(B) - P(A and B)

= 15/30 + 10/30 - 5/30 (see bonus visual below)

= 20/30

= 2/3.

000000000111111111122222222223

123456789012345678901234567890

-A-A-A-A-A-A-A-A-A-A-A-A-A-A-A 15 A's (multiples of 2)

--B--B--B--B--B--B--B--B--B--B 10 B's (multiples of 3)

-----C-----C-----C-----C-----C 5 C's (A and B) or (multiples of 6)

Answer 3

multiples of 2: 2, 4, 6, 8,10, 12, 14, 16, 18, 20 , 22, 24, 26, 28, 30

multiples of 3: 3, 6, 9, 18, 21, 24, 27 30

23 numbers, yet 4 are in both groups, so 19

19 / 30