Probability Group 2
1.* Two cards are randomly selected from an ordinary deck of 52 cards. What is the probability that they form a blackjack? That is, what is the probability that one of the cards is an ace and the other one either a ten, jack, queen, or king?
2.Urn A contains 3 red and 3 black marbles, whereas urn B contains 4 red and 6 black marbles. If a marble is randomly selected from each urn, what is the probability that the marbles will be of the same color?
3. There are 5 hotels in a certain town. If 3 people check into hotels in a day, what is the probability that they each check into a different hotel?
4.* A group of 6 men and 6 women is randomly divided into 2 groups of size 6 each. What is the probability that both groups will have the same number of men?
5.*Find the probability that 13 randomly selected cards from a standard 52 card deck will be missing at least one of the four suits.
6.* A pair of fair dice is rolled. What is the probability that the maximum of the two numbers is greater than 4?
7. What is the probability that four cards dealt at random from an ordinary 52 card deck will have one card from each suit?
8.Forty slips of paper each with a letter from the group A through E are put into a bag. The distribution of the letters in the bag is given in the graph:
If a slip of paper is selected at random, find the probability of
a) selecting an A.
b) selecting an A or an E.
c) not selecting a B.
d) not selecting a C or a D.
9.*A fair coin is tossed 17 times, find the probability of tossing at least 11 consecutive tails.
10.* A fair die is rolled five times. What is the probability that the number showing on each throw is higher than the one that was showing on the previous throw?
11.* George and Sam both plan to call Julie tonight to ask her for a date. George plans to call between 5:00 PM and 8:00 PM while Sam plans to call between 6:00 PM and 9:00 PM. If they pick their actual calling times at random, what is the probability that George calls before Sam?
12.* Two people agree to play the following game: they alternately draw marbles without replacement from an urn containing 4 white and 5 black marbles. Whoever removes the first white marble is the winner. What is the probability that the player who goes first will win the game?
13.*A box contains 5 sticks measuring 15, 30, 40, 60, and 90 centimeters in length. If three of the sticks are randomly chosen, what is the probability that they can be arranged to forma triangle?
14.* If there are 15 people in a room, what is the probability that at least two have the same birthday?
15. Six playing cards are lying face down on a table. You have been told that exactly two of the cards are Kings, but you don’t know which two. You pick two cards at random and turn them over. Which is more likely:
There will be at least one King among the two cards or there will be no King among the two cards?
16.* A confused secretary stuffs twelve checks into twelve envelopes and seals them. Then he realizes that he paid absolutely no attention to which check went into which envelope. Nevertheless, he goes ahead and mails them. What is the probability that
a) exactly one check is in its correct envelope?
b) exactly two checks are placed correctly?
c) exactly three checks are placed correctly?
d) none of the checks is placed correctly?
17. If the following spinner is spun, determine the probabilities of it landing on each possibility.
18. If a dart is thrown at random at this square dart board and hits the board, determine the probabilities of it landing on each possibility.
A / BB / A / A
A / B
B / A / B
19. If a dart is thrown at random at this circular dart board and hits the board, determine the probabilities of it landing on each possibility.
20.*Two fair dice consist of the following: one called the odd die is labeled with the numbers 1, 1, 3, 3, 5, and 5 and another called the even die is labeled with the numbers 2, 2, 4, 4, 6, and 6. If the dice are rolled, find the possible sums and their probabilities.
21.* Two numbers are selected at random one by one without replacement from the six numbers 1, 2, 3, 4, 5, and 6. What is the probability that the minimum of the two numbers is less than 4?
22.*One student in a class of men and women is to be chosen to represent the class. Each student is equally likely to be chosen, and the probability that a man is chosen is the probability that a woman is chosen. What is the ratio of the number of men to the total number of men and women?
23. In studying the cause of power failures, the following data has been gathered:
5% are due to transformer damage
80% are due to line damage
1% involve both
Find the probability that a given power failure involves
a) line damage given that there is transformer damage
b) transformer damage given that there is line damage
c) transformer damage but not line damage
d) transformer damage given that there is no line damage
e) transformer damage or line damage
24. The probability that a unit of blood was donated by a paid donor is .67. If the donor was paid, the probability of contracting serum hepatitis from the unit is .0144. If the donor was not paid, this probability is .0012. A patient receives a unit of blood. What is the probability of the patient contracting serum hepatitis from this source?
25. A chemical engineer is in charge of a particular process at an oil refinery. Past experience indicates that 10% of all shutdowns are due to equipment failure alone, 5% are due to a combination of equipment failure and operator error, and 40% involve operator error. A shutdown occurs. Find the probability that
a) equipment failure or operator error is involved
b) operator error alone is involved
c) neither operator nor equipment failure is involved
d) operator error is involved given that equipment failure occurs
e) operator error is involved given that equipment failure does not occur
26. A bowl contains 6 white and 9 red marbles. If 4 marbles are to be randomly selected without replacement, what is the probability that the first two selected are white and the last two are red?
27. A bowl initially contains 5 white and 7 black marbles. Each time a marble is selected, its color is noted and it is replaced in the bowl along with 2 other marbles of the same color.
a) Find the probability that the first two marbles selected are black and the next two white.
b) Find the probability that of the first 4 marbles selected, exactly two are black.
28. British and American spellings are rigour and rigor, respectively. A man staying at a Parisian hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40% of the English-speaking men at the hotel are British and 60% are American, what is the probability that the writer is British?
29.There are three coins in a box. One is a two-headed coin; another is a fair coin; and the third is a biased coin that comes up heads 75% of the time. When one of the three coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?
30. Die A has 4 red and 2 white faces, whereas die B has 2 red and 4 white faces. A fair coin is flipped once. If it lands on heads, the game continues with die A; if it lands on tails, then die B is to be used.
a) Find the probability of red on any throw.
b) If the first two throws are red, what is the probability of red on the third throw?
c) If red turns up on the first two throws, what is the probability that die A is being used?
31. The probability that a bus from Houston to Dallas will leave on time is .8, and the probability that it will leave on time and arrive on time is .72.
a) What is the probability that a bus arrives on time given that it leaves on time?
b) If we also know that the probability that a bus will arrive on time is .75, then what is the probability that a bus leaves on time given that it arrives on time?
32. A company uses a test for drug use that is 98% accurate – that is, it correctly identifies a person as a drug user or nonuser with probability .98 – and to reduce the chance of error, each job applicant is required to take two tests. If the outcomes of the two tests are independent events, what are the probabilities of these events?
a) A nonuser fails both tests.
b) A drug user is detected( fails at least one test)
c) A drug user passes both test.
33*. A parachutist will jump from an airplane and land in a square field that is 2 kilometers on each side. In each corner of the field there is a large tree. The parachutist’s ropes will get tangled in a tree if he lands within kilometer of its trunk. What is the probability that the parachutist will land in the field without getting caught in a tree?
34. Refer to the previous parachute problem. Suppose the parachutist gets caught in a corner tree if he lands within x kilometers of a tree. For what value of x will the probability of getting caught be 1?
35*. At a state fair a game is played by tossing a coin of radius 10 millimeters onto a large table ruled into congruent squares each with side measure of 30 millimeters. If the coin lands entirely within some square, the player wins a prize. If the coin touches or crosses the edge of any square, the player loses. Assuming that the coin lands somewhere on the table, what’s the probability that the player wins?
36. Refer to the previous problem. What should be the radius of the coin so that the probability of wining is .5?