Basic Counting Principles
How many ways can you arrange 3 books on a shelf?
3!
What is the probability of rolling an even number on a fair six-sided die?
3/6 = 1/2
If P(A) = 0.4 and P(B) = 0.5, and A and B are independent, what is P(A and B)?
0.4 * 0.5 = 0.2
How many integers between 1 and 100 are divisible by 2?
50
What is the expected value of rolling a fair six-sided die?
3.5
How many ways can you choose 2 toppings out of 5 for a pizza?
5 choose 2
What is the probability of drawing a heart from a standard deck of 52 cards?
13/52 = 1/4
If P(A) = 0.6, P(B) = 0.3, and P(A|B) = 0.4, what is P(A and B)?
P(A|B) × P(B) = 0.4 × 0.3 = 0.12
How many integers between 1 and 100 are divisible by 2 or 3?
50 + 33 - 16 = 67
What is the expected number of heads when flipping a fair coin 3 times?
3 × 0.5 = 1.5
How many 4-digit PINs can be formed if no digit is repeated?
10 × 9 × 8 × 7 = 5040
What is the probability of rolling a sum of 7 with two fair dice?
1/6
A box contains 3 red balls and 2 blue balls. If you draw two balls without replacement, what is the probability that both are red?
(3/5) × (2/4) = 3/10
How many ways can you arrange 5 people in a line if two specific people must stand next to each other?
2 × 4! = 48
A game costs 5 to play. You roll and ieandwin5toplay.Yourolladieandwin10 if you roll a 6, otherwise you win nothing. What is the expected value of the game?
(1/6 × 10) + (5/6 × 0) - 5 = -3.33
How many ways can you arrange the letters in the word 'MISSISSIPPI'?
11! / (4! × 4! × 2!) = 34,650
What is the probability of flipping exactly 3 heads in 5 flips of a fair coin?
5 choose 3 / (2 ^ 5) = 5 / 16
A disease affects 1% of the population. A test for the disease is 99% accurate. What is the probability that a person has the disease given they tested positive?
P(Disease|Positive) = (0.99 × 0.01) / [(0.99 × 0.01) + (0.01 × 0.99)] = 0.5
How many ways can you distribute 10 identical candies to 4 children if some children may get no candies?
C(10 + 4 - 1, 4 - 1) = C(13,3) = 286
What is the variance of the number of heads when flipping a fair coin 10 times?
10 × 0.5 × 0.5 = 2.5
How many ways can you distribute 10 identical candies to 4 children if each child must get at least one candy?
C(9,3) = 84
What is the probability of drawing two aces in a row from a standard deck without replacement?
(4 / 52) * (3/51) = 1 / 221
In a school, 60% of students play soccer, 40% play basketball, and 20% play both. What is the probability that a student plays soccer given they play basketball?
P(Soccer|Basketball) = P(Soccer and Basketball) / P(Basketball) = 0.2 / 0.4 = 0.5
How many ways can you partition a set of 4 distinct objects into 2 non-empty subsets?
2 ^ 14 - 2 = 14
You roll two dice. What is the expected value of the product of the two numbers rolled?
E[X] × E[Y] = 3.5 × 3.5 = 12.25