Practice (86)
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Avi and Hari agree to meet at their favorite restaurant between 5:00 p.m. and 6:00 p.m. They have agreed that the person who arrives first will wait for the other only 15 minutes before leaving. What is the probability that the two of them will actually meet at the restaurant, assuming that the arrival times are random within the hour? Express your answer as a common fraction.
Jean is twice as likely to make a free throw as she is to miss it. What is the probability that she will miss $3$ times in a row?
In one roll of four standard six-sided dice, what is the probability of rolling exactly three different numbers?
A bag contains ten each of red and yellow balls. The balls of each color are numbered from 1 to 10. If two balls are drawn at random, without replacement, then what is the probability that the yellow ball numbered 3 is drawn followed by a red ball?
A $5 \times 5 \times 5$ cube is painted on 5 of its 6 faces. It is then cut into 125 unit cubes. One unit cube is randomly selected and rolled. We are asked to find the probability that the top face of the cube that is rolled is painted.
If the letters of the word ELEMENT are randomly arranged, what is the probability that the three E's are consecutive?
Bessie shuffles a standard 52-card deck and draws five cards without replacement. She notices that all five of the cards she drew are red. If she draws one more card from the remaining cards in the deck, what is the probability that she draws another red card?
Among all pairs of real numbers $(x, y)$ such that $\sin\sin x=\sin\sin y$ with $-10\pi \le x, y \le 10\pi$. Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.
An ant begins at a vertex of a convex regular icosahedron (a figure with 20 triangular faces and 12 vertices). The ant moves along one edge at a time. Each time, the ant reaches a vertex, it randomly choose to next walk along any of the edges extending from that vertex (including the edge it just arrived from). Find the probability that after walking along exactly six (not necessarily distinct) edges, the ant finds itself at its starting vertex.
Ten unfair coins with probability of $1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{10}$ of showing heads are flipped. What is the probability that odd number of heads are shown?
Ten unfair coins with probability of $1, \frac{1}{3}, \frac{1}{4}, \dots, \frac{1}{11}$ of showing heads are flipped. What is the probability that odd number of heads are shown?
Let $a$ and $b$ be two randomly selected points on a line segment of unit length. What is the probability that their distance is not more than $\frac{1}{2}$?
Randomly select $3$ real numbers $x$, $y$, and $z$ between 0 and 1. What is the probability that $x^2 + y^2 + z^2 > 1$?
There are several equally spaced parallel lines on a table. The distance between two adjacent lines is $2a$. On the table, toss a coin with a radius of $r$, $(r < a)$. Find the probability that the coin does not touch any line.
Joe breaks a $10$-meter long stick into three shorter sticks. Find the probability that these three sticks can form a triangle.
Break a stick into two parts. What is the probability that the length of one part is at least twice of that of the other?
Two people agree to meet at a place some time in the next 10 days. They have also agreed whoever arrives the place should wait for the other for 3 days and then leave. What is the probability that they will see each other?
In the following diagram, $\overline{AO}= 2$, $\overline{BO} = 5$, and $\angle{AOB} = 60^\circ$. Point $C$ is selected on $\overline{BO}$ randomly. Find the probability that $\triangle{AOC}$ is an acute triangle.
A girl and a guy are going to arrive at a train station. If they arrive within $10$ minutes of each other, they will instantly fall in love and live happily ever after. But after $10$ minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between $8$ AM and $9$ AM with equal probability. The guy will arrive between $7$ AM and $8:30$ AM, also with equal probability. Find the probability that the probability that they fall in love.
Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
Mary plans to ask Joe to water the flowers during her summer vacation. Joe has a $10\%$ chance of forgetting this chore. If the flowers have an $85\%$ survival rate when watered but only a $20\%$ survival rate when not watered, what is the probability that the flowers will die upon Mary's return?
The germination rates of two different seeds are measured at $90\%$ and $80\%$, respectively. Find the probability that
- both will germinate
- at least one will germinate
- exactly one will germinate
A bug crawls from $A$ along a grid. It never goes backward, it crawls towards all the other possible directions with equal probability. For example:
- At $A$, it may crawl to either $B$ or $D$ with a 50-50 chance
- At $E$ (coming from $D$), it may crawl to $B$, $F$, or $H$ with a $\frac{1}{3}$ chance each
- At $C$ (coming from $B$), it will crawl to $F$ for sure
The questions are, from $A$:
- What is the probability of it landing at $E$ in 2 steps?
- What is the probability of it landing at $F$ in 3 steps?
- What is the probability of it landing at $G$ in 4 steps?
The probability that Alice can solve a given problem is $1/2$. Beth has $1/3$ chance to solve the same problem. Carol's chance to solve it is $1/4$. If all them work on this problem independently, what is the probability that one and only one of them solves it?
Let $a, b, c, m, n, p, k$ be positive real numbers that satisfy $a+m = b+n = c+p=k$. Show that $an+bp+cm < k^2$.