Math - EMF - Probability
A decimal die is tossed four times, thus producing a block of four random digits. The block is said to be an increasing block if it is of the type 1569, that is, if from the second digit on, each digit stands for a larger number than the previous digit does. What is the probability of obtaining an increasing block?
0.021
Let us suppose that the Sikinian coin tester is thinking about changing his criterion for rejecting coins. In each of the following cases, he plans to reject the coin being tested if the result of a sequence of tosses is as bad as or worse than the one shown. In each case calculate the probability that a fair coin will be rejected.
0.0215
Probability of having Two pairs (a pair and a second pair of a different digit). Digits(0-9). 4 numbers. BAd3
0.027
The right side of the passageway in the figure on the right is lit and the left side is dark. 16 rats have been sent down the maze and 12 have chosen the dark side. What is the probability that a preference for darkness as extreme as this or even more so results from pure chance?
0.038
In a certain game at the casino of Monte Bello you roll three 6-sided dice. If your total score is greater than or equal to 16 you win 20 kulotniks. If your score is less than 16 you lose 1 kulotnik. What is the probability of winning?
0.046
A newborn chicken is given some imitation grain made of paper in both round and triangular shapes of equal size and number. You observe the following pecking sequence: 🔴🔴🔴🔺🔺🔴🔴🔴🔴🔴 This makes you suspect that chickens prefer round objects that look like grain. But suppose chickens are indifferent to shape. What is the probability of obtaining by pure chance a result that is biased toward round shapes as much as this or even more so?
0.0547
Ex 75
0.1001
Let us suppose that the Sikinian coin tester is thinking about changing his criterion for rejecting coins. In each of the following cases, he plans to reject the coin being tested if the result of a sequence of tosses is as bad as or worse than the one shown. In each case calculate the probability that a fair coin will be rejected.
0.1460
The spinner in the figure on the right is spun seven times. What is the probability of obtaining at most two a's
0.2266
A symmetric coin is tossed nine times. What is the probability of obtaining five heads?
0.2461
A fair decimal die is rolled three times. What is the probability that The digit 1 occurs at least once?
0.271
The spinner on the right is spun eight times. (The spinner = 1/4 = a and 3/4 = b). Find the probability of obtaining a word consisting of two a's and six b's. If p is the probability of choosing such a word, give your answer in the form BAd3(p)?
0.311
A symmetric coin is tossed nine times. What is the probability of obtaining at least 5 heads?
0.5
A symmetric coin is tossed nine times. What is the probability of obtaining at most four heads?
0.5
Ex 68
0.5143
A machine produces yarn in such a way that the probability of any given meter of yarn being fault-free is 4/5. What is the probability that a 1/2-meter piece is fault-free? (Give the best D2 approximation for this part.)
0.89
A machine produces yarn in such a way that the probability of any given meter of yarn being fault-free is 4/5. What is the probability that at least one of two 1-meter pieces is fault-free?
0.96
Messages consisting of 0's and 1's are transmitted through a communication channel. Because of various disturbances, there is a probability of 0.1 that what is sent as a 0 will be received as a 1 and vice versa. You want to transmit an important message consisting of one digit. So, to protect it from corruption, you transmit 00000 instead of 0 and 11111 instead of 1. Meanwhile, the receiver uses majority decoding, which works as follows: Suppose the message you want to send is 0. That means you will transmit 00000. The set of all the messages that could possibly reach the receiver is M. If p is the probability that the message will be decoded correctly (using majority decoding), give your answer in the form BAd3(p).
0.991
A number ω is selected at random from the set Ω = {1,2,3,...,100}. Find the probability that ω is a square number (for example, 6 to the power of 2 or 9 to the power of 2)
1/10
A box contains two red and four yellow balls. Balls are drawn out of the box at random one at a time without replacement until two consecutive balls have the same color. Draw a tree of this experiment and find the probability that the experiment ends when the sixth ball is drawn.
1/15
A number ω is selected at random from the set Ω = {1,2,3,...,100}. Find the probability that ω is an even number
1/2
Suppose that when Richie plays chess against Jemima and Eve he beats Jemima with a probability of 1/3 and he beats Eve with a probability of 9/10. What is the probability that he will win two in a row if he plays in the order Jemima-Eve-Jemima?
1/2
Two dice are tossed. One is red and the other is blue. What is the probability that five dots are showing altogether on the two dice?
1/9
In how many ways can ten students be seated in a classroom containing ten seats? (No seat-sharing is allowed!)
10! = 3,628,800
A coach must reduce his basketball team from 13 players to 11 players. What is the probability that a particular player, Smith, will be chosen to remain on the team?
11/13
Two dice are tossed. One is red and the other is blue. If this pair of dice are tossed 1,000 times, estimate the number of times one should expect that the total number of dots showing will be five.
111
Three hunters each have probability 4/5 of hitting a target. If each hunter fires exactly one shot at the target, what is the probability that it will be hit at least once?
124/125
Ex 79
16/81
About 1000 years ago, when a thief was caught in Baghdad, he or she was thrown into a dark dungeon with three doors. One door led to freedom after one hour of travelling. The second door led into a tunnel which deposited the thief back in the dungeon after one days' travel. The third door led into a tunnel which returned to the dungeon after three days' travel. On being thrown into the dungeon each thief was given enough food and drink for four days and one hour. What is the probability of escape if doors are chosen at random? Note: If the poor prisoner went on a path that returned him to the dungeon, he couldn't tell which door(s) he had already visited and still had to choose one door at random from the three doors.
166/243
What are the possible number of license plates if they consist of three letters followed by three digits?
17,576,000
A box contains three red and seven blue balls. Three balls are drawn out of this box at random and without replacement. Find the probability of obtaining at least one red ball.
17/24
Consider the 11 letters of the word MISSISSIPPI. Two of these letters are selected at random, one by one without replacement. What is the probability that exactly one of the letters is P?
18/55
About 1000 years ago, when a thief was caught in Baghdad, he or she was thrown into a dark dungeon with three doors. One door led to freedom after one hour of travelling. The second door led into a tunnel which deposited the thief back in the dungeon after two days' travel. The third door led into a tunnel which returned to the dungeon after three days' travel. On being thrown into the dungeon each thief was given enough food and drink for five days and one hour. What is the probability of escape if doors are chosen at random? Note: If the poor prisoner went on a path that returned him to the dungeon, he couldn't tell which door(s) he had already visited and still had to choose one door at random from the three doors.
2/3
52 take 49
22,100
You have two red, three black, and five white balls. If any two balls of the same color are indistinguishable, how many distinct patterns can be made by lining them up from left to right?
2520
In how many distinct ways can the letters aaabbbbbcccc Be ordered
27,720
A true-false exam has five questions. Andy is completely ignorant and so he tosses a fair coin to decide his answer to each question. What is the probability that he scores at least four correct?
3/16
Carolyn and Arianna are playing a game in which a point is awarded to one of the players at each turn. The winner is the one whose score first reaches 3. If at each turn both Carolyn and Arianna have probability 1/2 of winning the point, what is the probability that the game lasts five turns?
3/8
Problem 3 review quiz(b)
3/8
A number ω is selected at random from the set Ω = {1,2,3,...,100}. Find the probability that ω is a multiple of 3
33/100
Suppose that when Richie plays chess against Jemima and Eve he beats Jemima with a probability of 1/3 and he beats Eve with a probability of 9/10. What is the probability that he will win two in a row if he plays in the order Eve-Jemima-Eve?
33/100
In how many distinguishable ways can the letters of the word ILLINOIS be ordered?
3360
In a restaurant there are two tables seating three and four guests, respectively. In how many ways can a party of seven guests be separated into a group of three and a group of four?
35
Seven students take a test. If each student has probability 1/2 of passing the test, what is the probability that exactly three of them will pass?
35/128
A box contains two red and four yellow balls. Balls are drawn out of the box at random one at a time without replacement until two consecutive balls have the same color. Draw a tree of this experiment and find the probability that the experiment ends when the third ball is drawn.
4/15
Ex 46
4/9
Arin and Calum hit a target with probabilities 4/5 and 7/10, respectively. They fire at the target simultaneously. Find the probability that at least one of them hits the target
47/50
There are 2 red balls and 2 green balls in box #1 and 3 red balls and 2 green balls in box #2 A ball is drawn at random out of box #1 and placed in box #2 without its color being observed. If a ball is then drawn out of box #2, what is the probability that it is green?
5/12
Five people whose probabilities of hitting a target are 1/2, 1/3, 1/4, 1/5, 1/6, respectively, shoot simultaneously at the target. What is the probability that the target will be hit at least once?
5/6
What are the possible number of license plates if they consist of two letters followed by four digits?
6,760,000
A box contains three green balls and four red balls. Two balls are drawn one by one out of this box without replacement. What is the probability that at least one of the two balls is red?
6/7
Ex 79
8/81
Two Sikinians, Messrs. Amba (A) and Bamba (B), each put 50 kulotniks into a pot. Then they made a sequence of spins with the fair spinner shown on the right. On each spin either A or B scored a point. The first one to reach a score of ten points was to win the pot. But unfortunately they were forced to terminate the game when A had eight points and B had six points. How should the stake money be split so as to favor the man A who is ahead? (To the nearest kulotnik.)
A = 81.25 B=18.75
In a certain game at the casino of Monte Bello you roll three 6-sided dice. If your total score is greater than or equal to 16 you win 20 kulotniks. If your score is less than 16 you lose 1 kulotnik. If you played the game 1000 times would you expect to be behind or ahead in terms of total winnings and losses, and (to the nearest kulotnik), by how much?
Behind by 28
The sum of the numbers corresponding to the two digits is less than 9
p(B) = 9/20
The first digit represents a larger number than the second digit
p(C) = 9/20
