Probability: all problems

Two 6-sided dice are rolled. What is the probability that the sum of the numbers rolled is 2?

Two 6-sided dice are rolled. What is the probability that the sum of the numbers rolled is 4?

Two dice are rolled. What is the probability that the sum of the numbers rolled is 6?

A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?

A fair 6 sided die is rolled twice. What is the probability that the first number that comes up is greater than the second number?

A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?

The arrows on the two spinners shown below are spun. Let the number $N$ equal 10 times the number on Spinner A, added to the number on Spinner B. What is the probability that $N$ is a perfect square number?

A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?

Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?

Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as

where $n$ is a positive integer. What is $n$?

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?

On a beach 50 people are wearing sunglasses and 35 people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is is also wearing sunglasses is $\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?

John has 9 red apples and 3 green apples. He randomly selects and eats an apple. Then, 3 of the red apples get rotten and are thrown away. From the remaining apples he randomly selects and eats another apple. What the probability that both of the selected apples were red is $\frac{a}{b}$, what is the value of $a + b$?

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?

A fair tetrahedral die, whose faces are numbered 1, 2, 3, and 4 is rolled three times. What is the probability that the sum of the numbers rolled is 7? Express your answer as a common fraction.

Two different numbers are randomly selected from the set $\{-2, -1, 0, 3, 4, 5\}$ and multiplied together. What is the probability that the product is 0?

Joe is standing at the point (0,0) in the infinite coordinate plane. On any given move, there is a $\frac{1}{3}$ chance of him moving up, $\frac{1}{3}$ chance of him moving right, $\frac{1}{6}$ chance of him moving down, and $\frac{1}{6}$ chance of him moving left. If the probability he will reach the point (3,3) in less than or equal to 8 moves can be described as $\frac{a}{3^b}$, then what is the value of $a + b$?

The Blazers and Cortzans are playing a series of 7 games. A team wins the series when they have won 4 games. Both teams have an equal likelihood of winning a game. If the probability the Cortzans win the series and lose less than 3 games can be expressed as $\frac{m}{n}$, find the value of $m + n$?

On the dart board shown in the figure below, the outer circle has radius 6 and the inner circle has radius 3. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd?

Eric is stacking twenty six Jenga Blocks. For $n \ge 1$, the probability that placing the $n$th block causes the whole tower to topple is $\frac{1}{26-n}$ assuming the tower still stands after placing $n - 1$ blocks. What is the expected number of blocks placed before the block that causes the tower to topple?

Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a 60% chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

A positive integer divisor of $11!$ is chosen at random. What is the probability that this divisor is prime? Express your answer as a common fraction.

Consider the regular octahedron as shown. Each edge of the octahedron has a length of 1. An ant starts at the vertex A and crawls a total distance of 3 units along the edges of the octahedron. Any time the ant reaches a vertex of the octahedron, it randomly chooses an edge to next crawl on that is different from the edge it just left. One such path the ant may take is shown. What is the probability that the ant will end up a distance greater than 1 from its starting point A ? Express your answer as a common fraction.