Test 1 Flashcards
Example 1. In a certain state, license plates have six characters that mayinclude letters and numerals. How many different license plates can be pro-duced if:
(a) Letters and numerals can be repeated?
(b) Each letter and numeral can be used at most once?
(c) The license plate must have a letter as its first character and eachletter or numeral can be used at most once?
(a) 2,176,782,336
(b) 1,402,410,240
(c) 1,012,851,840
Example 2. A teacher has 24 students in a classroom. For a group project, he decides to divide the students into four groups A, B, C, and D, of equal size. In how many ways can this be done?
2.31∗1012
Example 3. A package of eight light bulbs contains three defective lightbulbs. If two bulbs are randomly selected for use, find the probability that neither one is defective.
0.357
Example 4. A student is given a true-false test with ten questions. If shegets seven or more correct, she passes. Suppose that she guesses for eachquestion.
(a) What is the probability that she passes the test?
(b) What is the probability that she fails the test?
(a) 0.172
(b) 0.828
Example 5. Six employees of a firm are ranked from 1 to 6 in their abili-ties to fix problems with desktop computers. Three of these employees arerandomly selected to service three desktop computers. If all possible choicesof three out of the six are equally likely, find the probabilities for each of thefollowing events:
(a) The employee ranked number 1 is selected.
(b) The bottom three employees(4, 5, and 6) are selected.
(c) The highest-ranked employee among those selected has rank 3 or lower.
- 50
- 05
- 20
- 20
Example 6. In poker, each player is dealt five cards. What is the proba-bility of obtaining the following on the initial deal?
(a) Royal flush (ace, king, queen, jack, ten, all of the same suit)
(b) Straight flush (five cards in sequence all of the same suit)
(c) Four of a kind (four cards of the same rank)
(d) Full house (three cards of one rank, two cards of another rank)
(e) Three of a kind (three cards of the same rank, plus two other cardswhich are of different ranks)
(f) Two pairs (two sets of two cards of the same rank)
(g) One pair (two cards of equal rank)
4/2,598,960 36/2,598,960 624/2,598,960 3,744/2,598,960 54,912/2,598,960 123,552/2,598,960 1,098,240/2,598,960
A combination lock was left at a fitness center. The correct combination is a three-digit number d1d2d3, where di, i = 1, 2, 3, is selected from 0, 1, 2, 3,…, 9. How many different lock combinations are possible with such a lock?
1000
How many different license plates are possible if a state uses
(a) Two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)?
(b) Three letters followed by a three-digit integer? (In practice, it is possible that certain “spellings” are ruled out.)
(a) 6,760,000; (b) 17,576,000
How many four-letter code words are possible using the letters in IOWA if
(a) The letters may not be repeated?
(b) The letters may be repeated?
(a) 24; (b) 256
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
(a) 6, 7, 8, 9.
(b) 6, 7, 8, 8.
(c) 7, 7, 8, 8.
(d) 7, 8, 8, 8.
(a) 0.0024; (b) 0.0012; (c) 0.0006; (d) 0.0004
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
(a) Four games?
(b) Five games?
(c) Six games?
(d) Seven games?
(a) 2; (b) 8; (c) 20; (d) 40
Three students (S) and six faculty members (F) are on a panel discussing a new college policy.
(a) In how many different ways can the nine participants be lined up at a table in the front of the auditorium?
(b) How many lineups are possible, considering only the labels S and F?
(c) For each of the nine participants, you are to decide whether the participant did a good job or a poor job stating his or her opinion of the new policy; that is, give each of the nine participants a grade of G or P. How many different “scorecards” are possible?
(a) 362,880; (b) 84; (c) 512
A bridge hand is found by taking 13 cards at random and without replacement from a deck of 52 play-ing cards. Find the probability of drawing each of the following hands.
(a) One in which there are five spades, four hearts, three diamonds, and one club.
(b) One in which there are five spades, four hearts, two diamonds, and two clubs.
(c) One in which there are five spades, four hearts, one diamond, and three clubs.
(d) Suppose you are dealt five cards of one suit, four cards of another. Would the probability of having the other suits split 3 and 1 be greater than the probability of having them split 2 and 2?
(a) 0.00539; (b) 0.00882; (c) 0.00539; (d) Yes.
?
- 2917
0. 00622
Example 1. Consider flipping a coin four times. Let the random variable X denote the number of times heads was obtained.
(a) Find the pmf function of X.
(b) Sketch the histogram for this pmf.
(c) Find the cdf function of X.
(d) FindE(X)
(e) Find Var(X) and σ(X).
Lesson B
Example 2. Consider rolling two three-sided dice. Let Die A has the numbers 1, 1, 2. Let Die B has the numbers 1, 2, 3. Let the random variable X denote the total. (a) Find the pmf function of X. (b) Sketch the histogram for this pmf (c) Find the cdf function of X. (d) Find E(X). (e) Find Var(X) and σ(X).
Lesson B
Example 3. Letf(x) = (4−x)/c for x= 0,1,2,3.
(a) Find c so that f(x) is a pmf.
(b) Sketch the histogram for this pmf.
(c) Find the cdf function of X.
(d) FindE(X).
(e) Find Var(X) and σ(X).
(f) Suppose this function represents the number of hours performing a task. Suppose that the base pay if $50. You also get an additional $30 per
hour for each of the first two hours worked, and an additional $40 if the third hour is worked. What is your expected payment? What is the variance and standard deviation?
Lesson B
Example 4. PMF (X is the total of dice, (1, 1, 2) and (1, 2, 3) Using the dice from Example 2, suppose that you lose $2if the total of the two dice is 2, lose $1 if the total of the two dice is 3, win$1 if the total is 4, and win $2 if the total is 5.
(a) What are the expected winnings? Is this a good game?
(b) Compute the variance and standard deviation.
Lesson B
Example 5. Using the dice from Example 2, suppose that you lose $10if the total of the two dice is 2, lose $5 if the total of the two dice is 3, win$5 if the total is 4, and win $18 if the total is 5.
(a) What are the expected winnings? Is this a good game?
(b) Compute the variance and standard deviation.
Lesson B
Let the pmf of X be defined by f(x) = x/9, x = 2, 3, 4.
(a) Draw a bar graph for this pmf.
(b) Draw a probability histogram for this pmf
2.1
For each of the following, determine the constant c so that f(x) satisfies the conditions of being a pmf for a random variable X, and then depict each pmf as a bar graph:
(a) f(x) = x/c, x = 1, 2, 3, 4.
(b) f(x) = cx, x = 1, 2, 3,…, 10
(d) f(x) = c(x + 1)^2, x = 0, 1, 2, 3.
(a) 10; (b) 1/55; (d) 1/30;
The pmf of X is f(x) = (5 − x)/10, x = 1, 2, 3, 4. (a) Graph the pmf as a bar graph.
2.1
Let a random experiment be the casting of a pair of fair six-sided dice and let X equal the smaller of the out-comes if they are different and the common value if they are equal.
(a) With reasonable assumptions, find the pmf of X.
(b) Draw a probability histogram of the pmf of X.
(c) Let Y equal the range of the two outcomes (i.e., the absolute value of the difference of the largest and the smallest outcomes). Determine the pmf g(y)of Y for y = 0, 1, 2, 3, 4, 5.
(d) Draw a probability histogram for g(y)
2.1-7
Let the pmf of X be defined by f(x) = (1+|x−3|)/ 11, x = 1, 2, 3, 4, 5. Graph the pmf of X as a bar graph
2.1
Find E(X)
(a) f(x) = x/10, x = 1, 2, 3, 4.
(b) f(x) = (1/55)x, x = 1, 2, 3,…, 10
(d) f(x) = (1/30)(x + 1)^2, x = 0, 1, 2, 3.
(a) 3; (b) 7; (d) 7/3;
Let the random variable X be the number of days that a certain patient needs to be in the hospital. Suppose X has the pmf
f(x) = (5 − x)/10 , x = 1, 2, 3, 4
If the patient is to receive $200 from an insurance company for each of the first two days in the hospital and $100 for each day after the first two days, what is the expected payment for the hospitalization?
$360
In the gambling game chuck-a-luck, for a $1 bet it is possible to win $1, $2, or $3 with respective probabilities 75/216, 15/216, and 1/216. One dollar is lost with probability 125/216. Let X equal the payoff for this game and find E(X). Note that when a bet is won, the $1 that was bet, in addition to the $1, $2, or $3 that is won, is returned to the bettor
E(X) =−17/216 =−$0.0787
A roulette wheel used in an American casino has 38 slots, of which 18 are red, 18 are black, and two are green. A roulette wheel used in a French casino has 37 slots, of which 18 are red, 18 are black, and one is green. A ball is rolled around the wheel and ends up in one of the slots with equal probability. Suppose that a player bets on red. If a $1 bet is placed, the player wins $1 if the ball ends up in a red slot. (The player’s $1 bet is returned.) If the ball ends up in a black or green slot, the player loses $1. Find the expected value of this game to the player in
(a) The United States.
(b) France.
(a) −$1/19; (b) −$1/37.
