Probability - Problem Solving

If from each of the three boxes containing $3$ white and $1$ black, $2$ white and $2$ black, $1$ white and $3$ black balls, one ball is drawn at random. Then the probability that $2$ white and $1$ black balls will be drawn is?

If a fair 6-sided dice is rolled 3 times, what is the probability that we will get at least 1 even number and at least 1 odd number?

Three fair cubical dice are thrown. If the probability that the product of the scores on the three dice is $90$ is $\dfrac{a}{b}$, where $a,b$ are positive coprime integers, then find the value of $(b-a)$.

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Two players, Nihar and I, are playing a game in which we alternate tossing a fair coin and the first player to get a head wins. Given that I toss first, the probability that Nihar wins the game is $\dfrac{\alpha}{\beta}$, where $\alpha$ and $\beta$ are coprime positive integers.

Find $\alpha + \beta$.

Suppose you and 5 of your friends each brought a hat to a party. The hats are then put into a large box for a random-hat-draw. What is the probability that nobody selects his or her own hat?

A bag contains blue and green marbles. If 5 green marbles are removed from the bag, the probability of drawing a green marble from the remaining marbles would be 75/83 . If instead 7 blue marbles are added to the bag, the probability of drawing a blue marble would be 3/19 . What was the number of blue marbles in the bag before any changes were made?

Bob wants to keep a good-streak on Brilliant, so he logs in each day to Brilliant in the month of June. But he doesn't have much time, so he selects the first problem he sees, answers it randomly and logs out, despite whether it is correct or incorrect.

Assume that Bob answers all problems with $\frac{7}{13}$ probability of being correct. He gets only 10 problems correct, surprisingly in a row, out of the 30 he solves. If the probability that happens is $\frac{p}{q}$, where $p$ and $q$ are coprime positive integers, find the last $3$ digits of $p+q$.

Out of 10001 tickets numbered consecutively, 3 are drawn at random .

Find the chance that the numbers on them are in Arithmetic Progression .

The answer is of the form $\frac{l}{k}$ .

Find $k - l$ where $k$ and $l$ are co-prime integers.

HINT : You might consider solving for $2n + 1$ tickets .

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A bag contains a blue ball, some red balls, and some green balls.
You reach into​ the bag and pull out three balls at random.
The probability you pull out one of each color is exactly 3%.
How many balls were initially in the bag?

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Amanda decides to practice shooting hoops from the free throw line. She decides to take 100 shots before dinner.

Her first shot has a 50% chance of going in.

But for Amanda, every time she makes a shot, it builds her confidence, so the probability of making the next shot goes up, But every time she misses, she gets discouraged so the probability of her making her next shot goes down.

In fact, after $n$ shots, the probability of her making her next shot is given by $P = \dfrac{b+1}{n+2}$, where $b$ is the number of shots she has made so far (as opposed to ones she has missed).

So, after she has completed 100 shots, if the probability she has made exactly 83 of them is $\dfrac ab$, where $a$ and $b$ are coprime positive integers, what is $a+b$?

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