Probability and Counting Principles
Using counting methods to find probabilities
Probability and Counting Principles
Using counting methods to find probabilities
Math - Grade 9-12
- 1
A restaurant offers 4 choices of sandwich, 3 choices of side, and 5 choices of drink. How many different meal combinations are possible if you choose one of each?
Use the fundamental counting principle and multiply the number of choices for each category.
There are 60 different meal combinations possible because 4 multiplied by 3 multiplied by 5 equals 60. - 2
A locker code uses 3 different digits chosen from 0 through 9, and order matters. How many possible locker codes are there?
There are 720 possible locker codes because there are 10 choices for the first digit, 9 for the second, and 8 for the third, so 10 x 9 x 8 = 720. - 3
How many ways can 5 different books be arranged on a shelf?
This is a permutation of all 5 books.
The 5 books can be arranged in 120 ways because 5! = 5 x 4 x 3 x 2 x 1 = 120. - 4
A class has 12 students. In how many ways can a committee of 3 students be chosen?
A committee of 3 students can be chosen in 220 ways because order does not matter, so the number of combinations is 12C3 = 220. - 5
A bag contains 6 red marbles, 4 blue marbles, and 5 green marbles. What is the probability of drawing a blue marble at random?
Probability equals favorable outcomes divided by total outcomes.
The probability of drawing a blue marble is 4/15 because there are 4 blue marbles out of 15 total marbles. - 6
A standard deck of 52 cards is shuffled. What is the probability of drawing a king?
The probability of drawing a king is 1/13 because there are 4 kings in a deck of 52 cards, and 4/52 simplifies to 1/13. - 7
Two fair coins are flipped. What is the probability of getting exactly one head?
List all possible outcomes for two coin flips.
The probability of getting exactly one head is 1/2 because the outcomes HT and TH are favorable, and there are 4 equally likely outcomes total. - 8
A number cube is rolled twice. What is the probability that the sum is 7?
The probability that the sum is 7 is 1/6 because there are 6 favorable outcomes, (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), out of 36 total outcomes. - 9
A student has 7 shirts and 4 pairs of pants. How many outfits can the student make by choosing 1 shirt and 1 pair of pants?
Multiply the number of shirt choices by the number of pants choices.
The student can make 28 outfits because 7 x 4 = 28. - 10
How many different 4-letter arrangements can be made from the letters in the word MATH if each letter is used once?
There are 24 different arrangements because the 4 distinct letters can be arranged in 4! = 24 ways. - 11
From a group of 10 runners, how many ways can first place, second place, and third place be awarded?
Since first, second, and third are different positions, use permutations.
The places can be awarded in 720 ways because order matters, so the number of permutations is 10P3 = 10 x 9 x 8 = 720. - 12
A jar contains 3 yellow, 2 purple, and 5 orange candies. What is the probability of drawing an orange candy?
The probability of drawing an orange candy is 1/2 because there are 5 orange candies out of 10 total candies, and 5/10 simplifies to 1/2. - 13
A password is made using 2 letters followed by 2 digits. There are 26 letters and 10 digits, and repetition is allowed. How many passwords are possible?
Treat each position in the password as a separate choice and multiply.
There are 67,600 possible passwords because 26 x 26 x 10 x 10 = 67,600. - 14
What is the probability of choosing a vowel at random from the letters in the word EQUATION?
The probability of choosing a vowel is 5/8 because the letters are E, Q, U, A, T, I, O, N, and 5 of those 8 letters are vowels. - 15
A box contains 9 light bulbs, and 2 are defective. If one bulb is chosen at random, what is the probability that it is not defective?
Find how many bulbs are good, then divide by the total number of bulbs.
The probability that the bulb is not defective is 7/9 because 7 of the 9 bulbs are not defective.