Probability Stretch1. An integer is chosen at random from the set of all positive integers. What 1.

is the probability that the number is divisible by 3 and by 7? Express youranswer as a common fraction.

2. Four fair coins are flipped. What is the probability that at least two of 2.them will show heads? Express your answer as a common fraction.

3. A woman flips two fair coins until at least one of them shows heads. 3.Determine the probability that they both show heads. Express your answeras a common fraction.

4. Robyn selected three cards from a standard deck of 52 playing cards. 4.Amazingly, all three were aces! She set those three cards aside and thenchose one from the cards remaining in the deck. What is the probabilitythat it is a red ace? Express your answer as a common fraction.

5. The probability that Shaquille makes a free throw in a basketball game is 5.0.60. In a one-and-one situation, he shoots a second free throw only if hemakes the first. Find the probability that Shaquille will make both basketswhen he shoots one-and-one. Express your answer as a decimal to thehundredths.

6. Students at Wooden High School go to school 180 days a year. Ms. Hines, 6.the geometry teacher, assigns homework 108 days a year. Mr. Chien, thebiology teacher, assigns homework 105 days a year. On a randomly selectedday, what is the probability that a student in Ms. Hines's geometry classand in Mr. Chien's biology class will not have homework in either class?Express your answer as a common fraction.

7. Calculate the probability that • 7.three points chosen at random from theten equally-spaced points shown form an • •equilateral triangle. Express your answeras a common fraction. • • •

. ::' • • • •

8. On a canoe trip, the guide decides which two people will ride together in 8.a canoe. There are 12 people on the trip. Compute the probability thatKristen is assigned the same canoe as her best friend Karen. Express youranswer as a common fraction.

9. Two positive integers are randomly chosen. Determine the probability that 9.their product is even. Express your answer as a common fraction.

10. At a factory, a machine puts a cap and a label on a bottle. The caps, 10.labels and bottles are each available in the same four colors. Calculatethe prohA,hility that the cap, label and hot.t.le will all be different colors.Express your answer db a, common fraction.

IdATHCOUNTS lDD7-D8

Probability Stretch1. A bag contains 3 red and 5 blue marbles. If one marble is 1.

randomly drawn from the bag, what is the probability that it isblue? Express your answer as a common fraction.

2. A pocket full of change consists of 5 nickels, 2 dimes and 1 quarter. 2.Two coins are randomly chosen from the bag without replacement.What is the probability that their combined value is 15¢? Expressyour answer as a common fraction.

3. Two fair coins are flipped. What is the probability that both show 3.heads? Express your answer as a common fraction.

4. Two of the digits 1-5 are selected at random without replacement. 4.What is the probability that the positive difference between thetwo numbers is 3? Express your answer as a common fraction.

5. The height of the shaded rectangle 5.shown is one-half the height ofthe square, and the width of therectangle is one-third the width ofthe square. If a point is chosen atrandom inside the square, what isthe probability that it is also insidethe rectangle? Express your answeras a cornmon fraction.

6. Chi-Bin selected a positive multiple of 7 less than 70. Chi-Kai 6.selected a positive multiple of 11 less than 70. What is theprobability that they selected the same integer?

7. Alex has a lo chance of making a free throw. What is the 7.probability that she will make both of her next two free throws?Express your answer as a cornman fraction.

8. Tiles numbered 1-6 are each placed randomly into one of 8.three different boxes. What is the probability that each boxcontains 2 tiles? Express your answer as a common fraction.

9. Matt is given a jar with eight marbles, six of which are red and 9.two of which are green. Matt then randomly draws marbleswithout replacement. If he selects a red marble, he puts it in thefirst box; similarly, he continues to place all red marbles he selectsinto the first box until a green marble is chosen. When a greenmarble is chosen, he puts the first box aside. He then selects andplaces red marbles in the second box until the second green marbleis chosen. When the second green marble is selected, he will putthe second box aside and place all remaining red marbles in athird box. After all marbles are drawn, what is the probabilitythat each box contains two red marbles? Express your answer as acommon fraction.

10. One letter' is randomly selected from the word MATH and 10.one letter is randomly selected from the word CO UNTS. What. isthe probability that both letters selected arc consonants? Expressyour answer as a common fraction.

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MATHCOUNTS 1998-99

Probability Stretch1. There are an equal number of pennies, nickels, dimes and quarters 1.

in a bag. What is the probability that the combined value offour coins randomly selected with replacement will be 41 cents?Express your answer as a common fraction. (Problem submittedby Stanley Levinson, P.E)

2. What is the probability that the last three digits of a randomly 2.selected phone number are all prime? Express your answer as acommon fraction.

3. A five-digit number is created using the digits 1-5 each once. 3.What is the probability that the number is odd? Express youranswer as a common fraction.

4. The digits 2, 3, 4, 7 and 8 are each used once to form a five-digit 4.number. What is the probability that the tens digit is odd andthe number is divisible by 4? Express your answer as a commonfraction.

5. The sum of the digits of a number is 9. What is the probability 5.that the number is prime?

6. Two fair coins are flipped. What is the probability that at least 6.one coin is heads? Express your answer as a common fraction.

7. In a bag are only red and white marbles. The probability of 7.choosing a red marble is §. The probability of choosing two redmarbles without replacement is }30• How many white marbles arein the bag? (Problem submitted by coach Will Murnane.)

8. Two coins are flipped until at least one of them is heads. What is 8.the probability that both of them are heads? Express your answeras a common fraction.

9. Six number cubes with faces numbered 1-6 are rolled. The sum of 9.the numbers on the top faces is calculated. What is the most likelysum? (Problem submitted by alumnus Matthew Mendocino.)

10. What is the probability that apoint chosen inside the largestrectangle is not within a shadedregion? Express your answer as acommon fraction.

2 2 2 2

2

MATH COUNTS 1OVV-2000

10.

10.

Probability StretchA drawer containseight red, eight yellow, eight green and eight black socks. What isthe probability of getting at least one pair of matching socks when five socks arerandomly pulled from the drawer?

Two cubes, each numbered with the integers 4 through 9, are tossed. What is theprobability that the product of the two numbers rolled is greater than 29 and lessthan 61? Express your answeras a common fraction.

Five of the 50 computers ina lab have a virus. Ten of thecomputers are selected at random. What is the probability thatnone of the selected computers are infected? Express your answeras a decimal to the nearest hundredth.

A positive odd integer less than 150 is randomlyselected. What is the probabilitythat the product of the number selected and the next consecutive integer is lessthan 10,000? Express your answer as a common fraction.

The probability that Shelley will get a hit in any at-bat is 0.3. What is theprobability that she will get exactly one hit in her next two at-bats? Express youranswer as a decimal to the nearest hundredth.

What is the probability that a randomly selected two-digit positive integer is aperfect square or a perfect cube? Express your answer as a common fraction.

A point is selected in the region bounded by \x-l\ <2, y<-x +l, and / >-2. Whatis the probability that the point is in quadrant IV? Express your answer as a commonfraction.

One-fourth of Matilda's candies are blue, 1/8 are green, 1/4 are yellow, and the restare red. Matilda selects one candy. What is the probability that the selected candyis red? Express your answer as a common fraction.

The figure shown is constructed of 16 congruent squares. What is the probabilitythat a randomly selected rectangle within the figure is a square? Express youranswer as a common fraction.

Positive integers from 1to 30 inclusive are selected without replacement. What isthe fewest numberof integers that must be selected to guarantee that the productof at least one pair is a perfect square?

Problem #5 submittedby mathleteHannah Richman, Manning,PA.

42 MATHCOUNTS 2000-01

MATHCOUNTS 2005-2006 71

Probability Stretch1. ________ What is the probability that a multiple of 2 is also a multiple of 3? Express your

answer as a common fraction.

2. ________ A bag contains eight white marbles, six red marbles and four blue marbles. How many blue marbles must be added to the bag so that the probability of choosing a blue marble is ½?

3. ________ Mary‛s class took two tests on Friday. Eighty percent of the class passed the math test, 90% of the class passed the English test, and 72% of the class passed both tests. What is the probability that a randomly selected student in Mary‛s class failed both tests? Express your answer as a percent.

4. ________ Matt will flip a fair coin four times. What is the probability that ½ or more of the flips will be Heads? Express your answer as a common fraction.

5. ________ John is about to play Aaron in the best-two-out-of-three chess tournament finals. If the probability that John wins an individual game against Aaron is 35%, what is the probability that John will win the tournament? Express your answer as a percent to the nearest tenth.

6. ________ Marcus and Al are playing Rock/Paper/Scissors. The first person to win 10 times does not have to do the dishes. The score is now 9 to 7 in favor of Marcus. If ties are ignored and not counted, what is the probability Marcus will not have to do dishes? Express your answer as a common fraction.

7. ________ If the numerator of a fraction is selected from {1, 3, 5, 7, 9} and the denominator of the fraction is selected from {2, 3, 4, 5, 6, 7}, what is the probability that the resulting fraction can be written as a terminating decimal? Express your answer as a common fraction.

8. ________ A six-sided die is weighted so that the probabilities of rolling a 1, 2, 3 or 4 are equal. The probabilities of rolling a 5 or 6 also are equal to each other. With this die, Kia is three times more likely to roll a 6 than a 2. What is the probability that she will roll a 4? Express your answer as a common fraction.

9. ________ Lisa has hidden prizes at the ends of some paths in the woods. However, some paths have no prizes at the end. At each choice in a path, her friends are equally likely to choose either direction since they can‛t see what is at the end of any path. According to the map of the paths and prizes, what is the probability of a friend getting to a prize? Express your answer as a common fraction.

10. _______ Marissa, Naomi, Saul, Bob and Carl want to represent their class on the student council. If their teacher randomly chooses two of these students, what are the odds she will choose the two girls? Express your answer in the form a :b . (Be careful to give the odds rather than the probability!)

marbles

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Start

PrizeNo Prize

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Prize

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MATHCOUNTS 2017-201812

Probability Stretch1. ___________

2. ___________

3. ___________

4. ___________

5. ___________

6. ___________

7. ___________

8. ___________

9. ___________

10. __________

Petra randomly selects a card from a standard deck of 52 playing cards. What is the percent probability that the card shows a red number greater than 6? Express your answer to the nearest hundredth.

Max has eight identical cups. Each cup contains a different combination of nickels, dimes and quarters, each totaling 45 cents. Max randomly selects a cup. What is the probability that the cup he selects contains at least three dimes? Express your answer as a common fraction.

A bag contains five chips numbered 2 through 6. Danya draws chips from the bag one at a time and sets them aside. After each draw, she totals the numbers on all the chips she has already drawn. What is the probability that at any point in this process her total will equal 10? Express your answer as a decimal to the nearest tenth.

A drawer contains five socks: two green and three blue. What is the probability that two socks pulled out of the drawer at random will match? Express your answer as a common fraction.

A penny, a nickel and a dime are flipped. What is the probability that at least two coins land heads up and one of them is the nickel? Express your answer as a common fraction.

When the circuit containing blinking lights A and B is turned on, lights A and B blink together. Then A blinks once every 5 seconds and B blinks once every 11 seconds. Lindsey looks at the two lights just in time to see A blink alone. What is the percent probability that the next light to blink will be A blinking alone?

What is the percent probability that a randomly selected multiple of 3 less than or equal to 3000 is also a multiple of 5?

Starting at the top and selecting paths randomly as you move downward, what is the probability of ending at an odd number? Express your answer as a common fraction.

A five-digit number is made by randomly ordering the digits 1, 2, 3, 4 and 5. What is the probability that this number is divisible by 4? Express your answer as a common fraction.

Pierre throws darts that land randomly in the dartboard shown here. The dartboard is a circle of radius 2 units, with an inner circle of radius 1 unit. Both circles are divided into six congruent sectors. What is the probability that a dart Pierre throws will land in one of the four inner numbered sectors? Express your answer as a decimal to the nearest hundredth.

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1 23

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8

1 2

3 4

MATHCOUNTS 2018-201912

Expected Value Stretch

11. __________

12. __________

13. __________

14. __________

An unfair six-sided die with faces labeled 1, 2, 3, 5, 8 and 13 is rolled. The table lists the probability of the die landing with each number showing on the top face. The expected value of the roll is the sum of the products of each face value and its corresponding probability of being rolled. What is the expected value when the die is rolled? Express your answer as a mixed number.

Terry plays a game with prizes of 5, 10, 15 and 20 dollars. The graph shows each possible prize amount and its corresponding probability. The expected value of her prize is the sum of the products of each prize and the probability of winning that prize. What is the expected value of Terry’s prize?

A fair 10-sided die with one face labeled 1, two faces labeled 2, three faces labeled 3 and four faces labeled 4 is rolled. What is the expected value when this die is rolled?

Ana has a bowl containing two square tiles, one with side length 2 cm and the other with side length 3 cm. She randomly chooses a tile from the bowl. The expected value of the area of the chosen tile is the sum of the products of each tile’s area and its corresponding probability of being chosen. If the probability of choosing a particular tile is proportional to its area, what is the expected value of the area of the tile Ana chooses? Express your answer as a common fraction.

1

2

3

5

8

13

Top Face Value Probability

13

1615

115

110

215

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0.5

0.4

0.3

0.2

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0.20

0.45

0.25

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Prize Amounts

cm2

E( X ) = p1 x1 + p2 x2 + p3 x3 + ⋯ + pn xn

If the outcomes of random variable X have values x1, x2, x3, ..., xn and the probabilities of these outcomes occurring are p1, p2, p3, ..., pn, respectively, then the expected value of the outcome is the sum of the products of the probability of each outcome and the value of that outcome.

MATHCOUNTS 2018-2019 13

15. __________

16. __________

17. __________

18. __________

19. __________

20. __________

For the dartboard shown, the number of points scored when a dart lands in each region is indicated. The innermost circle of the board has radius 1 inch, and each subsequent circle has a radius 2 inches greater than the previous circle. Kane throws a dart that lands randomly somewhere on the board. What is the expected value of the number of points he scores? Express your answer as a decimal to the nearest tenth.

Gwen randomly draws a card from a deck of 40 cards numbered 1 through 40. What is the expected value of the number on the card she draws? Express your answer as a decimal to the nearest tenth.

Luke paints each face of a 5 × 5 × 5 cube red. He then cuts the cube into 125 unit cubes and randomly chooses a single unit cube. What is the expected value of the number of painted faces on this unit cube? Express your answer as a decimal to the nearest tenth.

In each round of a particular game, Dinara can win at most one point. If she has a 70% chance of winning a point in each round, what is the expected value of Dinara’s total score after three rounds? Express your answer as a decimal to the nearest tenth.

Jo and her four friends each secretly pick a random integer from −5 to 5, inclusive. What is the expected value of the sum of the five chosen numbers?

Allen randomly distributes 1000 jelly beans into 10 jars lined up in a row from left to right. What is the expected value of the number of jelly beans in the leftmost jar?

100

10

50

25

points

faces

E( X + Y ) = E( X ) + E(Y )

A property of E is that it is a linear function of the random variable. So, for random variables X and Y, the expected value of the sum of random variables equals the sum of their expected values.

points

jellybeans