ECON 210 - Economic Statistics
Problem Set III
Part I - Theoretical Probability and Combinatorics
1) Craps is a gambling game in which the players make a wager on the outcome of a roll of a
a pair
of dice. The basic rules of the game, whether it be played in a casino or on the street, are quite
simple. Each player (called a shooter) gets turns rolling two dice, and calculating the value of the
roll as the sum of the face values. Each turn has two phases: “come out” and “point”. In order
to enter the game the shooter has to make a “come out” roll. A come out roll of value 2, 3, or 12
is called “craps” or “crapping out”, and the shooter loses the initial wager she made. A come out
roll of 7 or 11 is called a “natural” and the shooter wins the bet automatically. If the roll lands
on any other possible number, i.e. 4, 5, 6, 8, 9, 10 it establishes that value as the “point”. Once a
point has been established, the shooter must roll the same point value again in order to win. If she
rolls a 7 before she rolls her point, she is said to have “sevened out”, and loses the bet. Keeping in
mind that the game involves rolling
a pair
of dice, answer the following questions.
(i) How many events constitute the sample space of a roll of a pair of dice?
(ii) What is the probability that a shooter will “crap out” on the come out roll?
(iii) What is the probability that the shooter rolls a “natural”?
(iv) What is, then, the probability of establishing a “point”? (Hint: Remind yourself of the very
basic rules of probability we discussed and save yourself the trouble of doing too many calculations!)
2) Car number plates in California follow the pattern 1 ABC 234, i.e. the ﬁrst character is a number
(only from 1-9), the next three characters are letters and the last three are again numbers (from
0-9).
(i) How many unique number plates can be created of this pattern?
(ii) How many number plates can be created if no two characters on the plate were allowed to be
repeated?
3) In a random arrangement of the letters of the word VIOLENT, ﬁnd the probability that the
vowels occupy the even spots.
4) An urn
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has 8 black, 3 red and 9 white balls. If 3 balls were drawn from the urn at random,
ﬁnd the probability that: (i) all are black (ii) 2 are black and 1 is white, (iii) one is of each color,
(iv) none are red.
5) A pound has 60 German Shepherds and 40 Great Danes of both sexes. If they breed with each
other randomly, what is the probability that the puppies are: (i) pure breed German Shepherds,
(ii) pure breed Great Danes, (iii) German Shepherd-Great Danes? ]
6) Five digit numbers are being formed from the numbers 1, 2, 3, 4, 5; no digit being repeated.
Find the probability that a random number so formed is: (i) divisible by 5, (ii) divisible by 2, (iii)
greater than 23,000.
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An urn is a kind of a tall vase, usually used to store the ashes of a cremated person, but let’s not dwell on the
morbid stuﬀ.
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