Sihangli/scibench-rl huggingface dataset:
Problem: If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find An upper bound for $P(|X-33| \geq 14)$.
Answer:  $0.082$  
Numeric value: 0.082
Problem: Suppose that the length of life in hours (say, $X$ ) of a light bulb manufactured by company $A$ is $N(800,14400)$ and the length of life in hours (say, $Y$ ) of a light bulb manufactured by company $B$ is $N(850,2500)$. One bulb is randomly selected from each company and is burned until 'death.' Find the probability that the length of life of the bulb from company $A$ exceeds the length of life of the bulb from company $B$ by at least 15 hours.
Answer:  $0.3085$  
Numeric value: 0.3085
Problem: An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement.
Answer: $\frac{1}{16}$  
Numeric value: 0.0625
Problem:  If $P(A)=0.8, P(B)=0.5$, and $P(A \cup B)=0.9$. What is $P(A \cap B)$?
Answer:  0.9  
Numeric value: 0.9
Problem: Suppose that the alleles for eye color for a certain male fruit fly are $(R, W)$ and the alleles for eye color for the mating female fruit fly are $(R, W)$, where $R$ and $W$ represent red and white, respectively. Their offspring receive one allele for eye color from each parent. Assume that each of the four possible outcomes has equal probability. If an offspring ends up with either two white alleles or one red and one white allele for eye color, its eyes will look white. Given that an offspring's eyes look white, what is the conditional probability that it has two white alleles for eye color?
Answer: $\frac{1}{3}$  
Numeric value: 0.33333333
Problem: Consider the trial on which a 3 is first observed in successive rolls of a six-sided die. Let $A$ be the event that 3 is observed on the first trial. Let $B$ be the event that at least two trials are required to observe a 3 . Assuming that each side has probability $1 / 6$, find $P(A)$.
Answer: $\frac{1}{6}$  
Numeric value: 0.166666666
Problem: An urn contains four balls numbered 1 through 4 . The balls are selected one at a time without replacement. A match occurs if the ball numbered $m$ is the $m$ th ball selected. Let the event $A_i$ denote a match on the $i$ th draw, $i=1,2,3,4$. Extend this exercise so that there are $n$ balls in the urn. What is the limit of this probability as $n$ increases without bound?
Answer:  $1 - \frac{1}{e}$  
Numeric value: 0.6321205588
Problem:  Of a group of patients having injuries, $28 \%$ visit both a physical therapist and a chiropractor and $8 \%$ visit neither. Say that the probability of visiting a physical therapist exceeds the probability of visiting a chiropractor by $16 \%$. What is the probability of a randomly selected person from this group visiting a physical therapist?

Answer:  0.68  
Numeric value: 0.68
Problem: A doctor is concerned about the relationship between blood pressure and irregular heartbeats. Among her patients, she classifies blood pressures as high, normal, or low and heartbeats as regular or irregular and finds that 16\% have high blood pressure; (b) 19\% have low blood pressure; (c) $17 \%$ have an irregular heartbeat; (d) of those with an irregular heartbeat, $35 \%$ have high blood pressure; and (e) of those with normal blood pressure, $11 \%$ have an irregular heartbeat. What percentage of her patients have a regular heartbeat and low blood pressure?
Answer:  15.1 % 
Numeric value: 15.1
Problem: Roll a fair six-sided die three times. Let $A_1=$ $\{1$ or 2 on the first roll $\}, A_2=\{3$ or 4 on the second roll $\}$, and $A_3=\{5$ or 6 on the third roll $\}$. It is given that $P\left(A_i\right)=1 / 3, i=1,2,3 ; P\left(A_i \cap A_j\right)=(1 / 3)^2, i \neq j$; and $P\left(A_1 \cap A_2 \cap A_3\right)=(1 / 3)^3$. Use Theorem 1.1-6 to find $P\left(A_1 \cup A_2 \cup A_3\right)$.
Answer: $3(\frac{1}{3})-3(\frac{1}{3})^2+(\frac{1}{3})^3$  
Numeric value: 0.6296296296
Problem: Let $A$ and $B$ be independent events with $P(A)=$ $1 / 4$ and $P(B)=2 / 3$. Compute $P(A \cap B)$
Answer:  $\frac{1}{6}$  
Numeric value: 0.166666666
Problem: How many four-letter code words are possible using the letters in IOWA if the letters may not be repeated?
Answer:  24  
Numeric value: 24
Problem: A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, $d_1 d_2 d_3 d_4$, where $d_i, i=1,2,3,4$, is selected from $1,2,3,4,5,6,7$, and 8 . How many different lock combinations are possible with such a lock?
Answer:  4096  
Numeric value: 4096
Problem: An urn contains eight red and seven blue balls. A second urn contains an unknown number of red balls and nine blue balls. A ball is drawn from each urn at random, and the probability of getting two balls of the same color is $151 / 300$. How many red balls are in the second urn?
Answer:  11  
Numeric value: 11
Problem:  A typical roulette wheel used in a casino has 38 slots that are numbered $1,2,3, \ldots, 36,0,00$, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of $1 / 38$, and we are interested in the number of the slot into which the ball falls. Let $A=\{0,00\}$. Give the value of $P(A)$.
Answer: $\frac{2}{38}$ 
Numeric value: 0.0526315789
Problem: In the gambling game "craps," a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2,3 , or 12 . If the sum is $4,5,6$, 8,9 , or 10 , that number is called the bettor's "point." Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7 , the bettor wins. Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11 , $P(7$ or 11$)$.
Answer:  $\frac{8}{36}$  
Numeric value: 0.22222222
Problem: Given that $P(A \cup B)=0.76$ and $P\left(A \cup B^{\prime}\right)=0.87$, find $P(A)$.
Answer:  0.63  
Numeric value: 0.63
Problem: How many different license plates are possible if a state uses two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)?
Answer:  6760000  
Numeric value: 6760000
Problem: Let $A$ and $B$ be independent events with $P(A)=$ 0.7 and $P(B)=0.2$. Compute $P(A \cap B)$.

Answer:  0.14  
Numeric value: 0.14
Problem: Suppose that $A, B$, and $C$ are mutually independent events and that $P(A)=0.5, P(B)=0.8$, and $P(C)=$ 0.9 . Find the probabilities that all three events occur?
Answer:  0.36  
Numeric value: 0.36
Problem: A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of four of a kind (four cards of equal face value and one card of a different value).
Answer:  0.00024  
Numeric value: 0.00024
Problem: Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?
Answer:  362880  
Numeric value: 362880
Problem: Each of the 12 students in a class is given a fair 12 -sided die. In addition, each student is numbered from 1 to 12 . If the students roll their dice, what is the probability that there is at least one "match" (e.g., student 4 rolls a 4)?
Answer: $1-(11 / 12)^{12}$  
Numeric value: 0.648004372
Problem: The World Series in baseball continues until either the American League team or the National League team wins four games. How many different orders are possible (e.g., ANNAAA means the American League team wins in six games) if the series goes four games?
Answer:  2  
Numeric value: 2
Problem: Draw one card at random from a standard deck of cards. The sample space $S$ is the collection of the 52 cards. Assume that the probability set function assigns $1 / 52$ to each of the 52 outcomes. Let
$$
\begin{aligned}
A & =\{x: x \text { is a jack, queen, or king }\}, \\
B & =\{x: x \text { is a } 9,10, \text { or jack and } x \text { is red }\}, \\
C & =\{x: x \text { is a club }\}, \\
D & =\{x: x \text { is a diamond, a heart, or a spade }\} .
\end{aligned}
$$
Find $P(A)$
Answer: $\frac{12}{52}$  
Numeric value: 0.2307692308
Problem: An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacement, and you are told that at least one of them is orange. What is the probability that the other ball is also orange?
Answer: $\frac{1}{5}$  
Numeric value: 0.2
Problem: Bowl $B_1$ contains two white chips, bowl $B_2$ contains two red chips, bowl $B_3$ contains two white and two red chips, and bowl $B_4$ contains three white chips and one red chip. The probabilities of selecting bowl $B_1, B_2, B_3$, or $B_4$ are $1 / 2,1 / 4,1 / 8$, and $1 / 8$, respectively. A bowl is selected using these probabilities and a chip is then drawn at random. Find $P(W)$, the probability of drawing a white chip.
Answer:  $\frac{21}{32}$  
Numeric value: 0.65625
Problem: Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the longer segment is at least two times longer than the shorter segment.
Answer:  $\frac{2}{3}$  
Numeric value: 0.66666666666
Problem: In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select $6,7,8,9$.
Answer:  0.0024  
Numeric value:  0.0024
Problem: Suppose that a fair $n$-sided die is rolled $n$ independent times. A match occurs if side $i$ is observed on the $i$ th trial, $i=1,2, \ldots, n$. Find the limit of this probability as $n$ increases without bound.
Answer:  $ 1-1 / e$  
Numeric value: 0.6321205588