2. Probability · Mastering AMC 10/12

2.10%

Probability Basics

In this section you will be able to find a probability by counting equally likely outcomes: desired over total.

Roll two dice. What fraction of the 36 outcomes give an even sum?

Every probability in this chapter starts the same way. List the equally likely outcomes, count the ones you want, and divide by the total.

Look:the outcomes must be equally likely. “Sum is 2” and “sum is 7” are two sums, but 7 happens six times as often.

Exploreplay first, formulas later

Try this: Pick "Sum is even". Count the lit cells. What fraction of the 36 is that?
Try this: Why is the grid 6 × 6 and not 6 + 6? Think of every pair (first die, second die).
Try this: Press "Roll 1000 times". How close does the experiment get to the theory?

Name it

Definition 2.1.1 · Probability

Probability is the chance something occurs.

Theorem 2.1.2 · Probability

When all outcomes are equally likely,

\text{probability} = \frac{\text{Total number of Desired Outcomes}}{\text{Total Outcomes}}.

Watch itpredict, then reveal

Example 2.1

AIME

In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is $\frac{1}{n}$ , where $n$ is a positive integer. Find $n$ .

1
What must happen so that the same coloring appears after a $90^\circ$ rotation?

Predict first:
Click a choice or type its letter (A–E).

Try it

Problem 2.6.1

Two 6-sided dice are rolled. What is the probability that the sum of the two numbers is even?

Contest cornerreal contest problems

Problem 2.6.5

AMC 10

Two different numbers are selected at random from $\{1, 2, 3, 4, 5\}$ and multiplied together. What is the probability that the product is even?

Problem 2.6.8

AMC 10/12

The numbers $1, 2, \dots, 9$ are randomly placed into the 9 squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?

Problem 2.6.13

AMC 10

A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?

Checkpoint for section 2.1

Get 2 of 3 right to master this section. One try each, no hints.

2.20%

Distinguishability

In this section you will be able to decide whether to count with order or without, and spot cases that aren't equally likely.

Pick 2 different cards from cards numbered 1 to 10. Is the pair “3 then 7” different from “7 then 3”? For the probability, it doesn’t matter, as long as you’re consistent.

Counting ordered pairs multiplies the top and the bottom of the fraction by the same $2!$ , so it cancels. The trap is listing cases that are not equally likely, like the sums $2 + 2$ and $1 + 3$ .

Exploreplay first, formulas later

Pick 2 different cards from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Light: the product is odd.

{1, 2}
{1, 3}
{1, 4}
{1, 5}
{1, 6}
{1, 7}
{1, 8}
{1, 9}
{1, 10}
{2, 3}
{2, 4}
{2, 5}
{2, 6}
{2, 7}
{2, 8}
{2, 9}
{2, 10}
{3, 4}
{3, 5}
{3, 6}
{3, 7}
{3, 8}
{3, 9}
{3, 10}
{4, 5}
{4, 6}
{4, 7}
{4, 8}
{4, 9}
{4, 10}
{5, 6}
{5, 7}
{5, 8}
{5, 9}
{5, 10}
{6, 7}
{6, 8}
{6, 9}
{6, 10}
{7, 8}
{7, 9}
{7, 10}
{8, 9}
{8, 10}
{9, 10}

\frac{10}{45} = \frac{2}{9}

Counting order instead gives $\frac{20}{90} = \frac{2}{9}$ . Same answer!

Try this: Turn "Order matters" on. Both counts doubled. Did the probability change?
Try this: Press "Show me" on the second remark: two integers with sum 4. Why is the answer 1/3, not 1/2?
Try this: In sum-4 mode, which listed case stands for two orderings?

Name it

Remark 2.2.1

In probability problems, whether you decide to multiply for order or not is up to you, as both will give the same probability.

Remark 2.2.2

If you decide not to multiply for order, make sure the number of orderings is the same in every case. In Example 2.2, every pair had $2$ orderings. The same is true for groups of $5$ , which always have $5!$ orderings. But in Example 2.3, the two cases had different numbers of orderings, so leaving out order doesn't work there.

Watch itpredict, then reveal

Example 2.2

I have cards numbered from 1 to 10. What is the probability I pick a pair of 2 different cards that have an odd product?

1
What must happen for the cards to have an odd product?

Predict first:
Click a choice or type its letter (A–E).

Example 2.3

The sum of 2 positive integers is 4. Find the probability that one of the integers is a 2.

1
List the cases for the two integers.

Example 2.4

AIME

A deck of forty cards consists of four 1's, four 2's, ..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $\frac{m}{n}$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

1
After the pair is removed, how many cards are left?

Predict first:
Your prediction

open on YouTube

Example 2.5

AMC 10

Each of the 20 balls is tossed independently and at random into one of the 5 bins. Let $p$ be the probability that some bin ends up with 3 balls, another with 5 balls, and the other three with 4 balls each. Let $q$ be the probability that every bin ends up with 4 balls. What is $\frac{p}{q}$ ?

1
All $5^{20}$ tosses are equally likely, so $p$ and $q$ share a denominator. What do we compare instead?

open on YouTube

Contest cornerreal contest problems

Problem 2.6.4

AMC 10

Ms. Carr asks her students to read any 5 of the 10 books on a reading list. Harold randomly selects 5 books from this list, and Betty does the same. What is the probability that there are exactly 2 books that they both select?

Problem 2.6.9

AMC 10

Two cubical dice each have removable numbers 1 through 6. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is 7?

Problem 2.6.12

AMC 12

Let $S$ be the set of permutations of the sequence $1, 2, 3, 4, 5$ for which the first term is not $1$ . A permutation is chosen randomly from $S$ . The probability that the second term is $2$ , in lowest terms, is $\frac{a}{b}$ . What is $a + b$ ?

Problem 2.6.14

AMC 10

Forty slips are placed into a hat, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $p$ be the probability that all four slips bear the same number. Let $q$ be the probability that two of the slips bear a number $a$ and the other two bear a number $b \ne a$ . What is the value of $q/p$ ?

Checkpoint for section 2.2

Get 2 of 3 right to master this section. One try each, no hints.

2.30%

Probability of Independent Events

In this section you will be able to multiply the probabilities of independent events.

Two people roll two strange dice. One die has faces 1, 1, 2, 6, 15, 30 and the other 1, 2, 4, 18, 28, 44. How often is the product a multiple of 24?

Two events are independent when the result of one doesn’t change the chances of the other. Then you don’t need the whole grid: find each probability on its own and multiply.

Exploreplay first, formulas later

Try this: Pick "Product is a multiple of 24". Which faces of Person A's die can never work?
Try this: Look at the rows for 6 and 30. Is the chance for Person B the same in both rows?
Try this: Count: 2 good rows out of 6, and 3 good columns out of 6 in those rows. Does 1/3 × 1/2 match the grid?

Name it

Remark 2.3.1

To find the probability that two independent events both happen (the result of one doesn't depend on the other), find the probability of each event on its own and multiply them.

Independent dice: the chance that none of them shows a 6 is (5/6) × (5/6) × (5/6).

Watch itpredict, then reveal

Example 2.6

Omega Learn

Person A rolls a die with the numbers 1, 1, 2, 6, 15, 30 and Person B rolls a die with the numbers 1, 2, 4, 18, 28, 44. What is the probability that the product of the 2 numbers is a multiple of 24?

1
Factor $24$ . What does the product need?

open on YouTube

Example 2.7

What is the probability that the product of 4 dice rolls is a multiple of 648?

1
Factor $648$ .

open on YouTube

Contest cornerreal contest problems

Problem 2.6.6

AMC 10

A fair 6-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?

Problem 2.6.7

AMC 12

Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 is $\frac{1}{12}$ . What is the least possible number of faces on the two dice combined?

Problem 2.6.11

AMC 12

Tina randomly selects two distinct numbers from the set $\{1, 2, 3, 4, 5\}$ , and Sergio randomly selects a number from the set $\{1, 2, \dots, 10\}$ . What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?

Problem 2.6.15

AMC 10/12

A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \dots$ . What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

Problem 2.6.18

AMC 12

Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$ . What is the probability that Flora will eventually land at 10?

Checkpoint for section 2.3

Get 2 of 3 right to master this section. One try each, no hints.

2.40%

Probability of Dependent Events

In this section you will be able to use a probability tree when the first event changes the chances of the second.

There is a 40% chance of rain on Saturday and a 30% chance on Sunday. But rain on Saturday makes rain on Sunday twice as likely. So the days are not independent.

When events are dependent, multiply along a path of a tree: the first chance, times the second chance given the first happened. Then add the paths you want.

Look:the leaves of a tree always add up to 1. That makes complementary probability easy.

Exploreplay first, formulas later

Try this: Step x up until P(rain Sun) = 3/10. What is x?
Try this: Which leaves make "rain on at least one day"? Click them and add.
Try this: Compare P(rain Sun | rain Sat) with P(rain Sun). Are the two days independent?

Name it

Remark 2.4.1

When events are dependent, the probability that both $A$ and $B$ happen is

P(A \text{ and } B) = P(A) \times P(B \text{ given that } A \text{ happened}).

Watch itpredict, then reveal

Example 2.8

AIME

There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$ .

1
Let $x$ be the chance of rain on Sunday if it does not rain on Saturday. What is the chance if it does?

open on YouTube

Contest cornerreal contest problems

Problem 2.6.2

AMC 12

A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\frac{1}{3}$ . When 4 black cards are added to the deck, the probability of choosing red becomes $\frac{1}{4}$ . How many cards were in the deck originally?

Problem 2.6.3

AMC 10

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?

Problem 2.6.20(additional)

AIME

Jen enters a lottery by picking 4 distinct numbers from $S = \{1, 2, 3, \dots, 9, 10\}$ . 4 numbers are randomly chosen from $S$ . She wins a prize if at least two of her numbers were 2 of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of her winning the grand prize given that she won a prize is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Problem 2.6.21(additional)

AMC 10/12

An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?

Checkpoint for section 2.4

Get 2 of 3 right to master this section. One try each, no hints.

2.50%

Dependent or Independent Events?

In this section you will be able to decide whether one random choice changes the next, and count accordingly.

Mark gets a random seat in a 5 × 5 classroom, then Steve gets another. What is the chance Steve avoids Mark’s row and column?

Steve’s seat depends on Mark’s: Mark’s seat is no longer free. But here the number of bad seats is the same wherever Mark sits, so you can seat Mark first and just count Steve’s options.

Exploreplay first, formulas later

Try this: Move Mark to a corner. Does Steve's chance change?
Try this: Count the forbidden seats for any spot of Mark's. Why is it always 8?
Try this: Make the grid 3 × 3. Predict Steve's chance before you look.

Watch itpredict, then reveal

Example 2.9

Omega Learn

Mark must choose a seat from a $5 \times 5$ grid of chairs in the classroom. His enemy Steve does not like Mark, so he wants to choose a seat that is not in the same row or column as Mark. However, to his dismay, the teacher randomly assigns seats. What is the probability that Steve is not sitting in the same row or column as Mark?

1
Seat Mark first. How many seats are left for Steve?

Predict first:
Your prediction

open on YouTube

Checkpoint for section 2.5

Get 2 of 3 right to master this section. One try each, no hints.

2.6

Practice Problems

In this section you will be able to mix every tool from this chapter on contest problems.

Time to mix it all up: parity, symmetry, repeated rolls and infinite sums. Each problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 2.6.10

AMC 10/12

Each of 2023 balls is randomly placed into one of 3 bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?

Problem 2.6.16

AMC 12

This problem also appears in the book as 2.6.7.

Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 is $\frac{1}{12}$ . What is the least possible number of faces on the two dice combined?

Problem 2.6.17

AMC 10

This problem also appears in the book as 2.6.6.

A fair 6-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?

Problem 2.6.19

AMC 12Challenge

Three balls are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i = 1, 2, 3, \dots$ . More than one ball is allowed in each bin. The probability that the balls end up evenly spaced in distinct bins is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. (For example, the balls are evenly spaced if they are tossed into bins 3, 17, and 10.) What is $p + q$ ?