Probability (Day 1 and 2) – Blue Problems
Independent Events
1. There are 5 blue chips and 3 yellow chips in a bag. One chip is drawn from the bag.
The chip is placed back into the bag. A second chips is then drawn. What is the
probability that the two selected chips are of different colors? Express your
answer a common fraction.
2. There are two red balls and two white balls in a jar. One ball is drawn and replaced
with a ball of the other color. The jar is then shaken and one ball is chosen. What is
the probability that this ball is red? Express your answer as a common fraction.
3. A red marble, a blue marble and a yellow marble are placed in a bag. A marble is
drawn and not returned to the bag. A second marble is then drawn. What is the
probability that the first marble is blue and the second marble is yellow? Express
your answer a common fraction.
4. Two coins are each flipped one. What is
the probability of getting two heads?
Express your answer a common fraction.
5. Given that you randomly choose T or F for each answer on a ten question true-false
test, determine the probability that you will get all ten answers correct. Express
your answer a common fraction.
6. In the game of PIG, a player tosses a pair of standard dice and loses all of her
points if two 1’s are rolled. If the dice are rolled 144 times, how many times would
you expect to get a pair of 1’s?
7. Two standard dice are rolled. What is the probability that the product of the two
numbers rolled exceeds 8? Express your answer as a common fraction.
8. Aimee tosses one fair 6-sided die labeled 1 through 6 and one fair-4sided die
labeled 1 through 4. What is the probability that the sum Aimee rolls is less than 5?
Express your answer as a common fraction.
9. John, Kevin, Larry, Mary and Naomi all volunteered to do some math tutoring. If
their teacher randomly chooses two of the five students, what is the probability of
selecting the two girls? Express your answer as a common fraction.
10. Two standard 6-sided dice are tossed. What is the probability that the sum of the
two numbers rolled is greater than nine? Express your answer as a common fraction.
11. Chi-Bin selected a positive multiple of 7 less than 70. Chi-Kan selected a positive
multiple of 11 less than 70. What is the probability that they selected the same
integer?
12. Tara rolls three standard dice once. What is the probability that the sum of the
numbers rolled will be three or more? Express your answer as a common fraction.
13. What is the probability of tossing exactly two heads or exactly two tails when
three fair coins are tossed? Express your answer as a common fraction.
14. The probability that Shaquille makes a free throw in a basketball game is 0.60. In a
one-and-one situation, he shoots a second free throw only if he makes the first.
Find the probability that Shaquille will make both baskets when he shoots one-andone. Express your answer as a decimal to the hundredths.
Dependent Events
1. Two cards are randomly selected without replacement from a set of four cards
numbered 2, 3, 4 and 5. What is the probability that the sum of the numbers on the
two cards selected is 7? Express your answer as a common fraction.
2. In the game of Math Cards, a Magic Prime hand must have four cards, each with a
value of either a 2, 3, 5, 7, Jack or King, but no more than one card with the same
value. If our cards are randomly chosen without replacement from a standard deck
of 52 cards, what is the probability of getting a magic prime hand? Express your
answer as a decimal to the nearest thousandth.
3. A bag contains five red socks and eight blue socks. Lucky reaches into the bag and
randomly selects two socks without replacement. What is the probability that Lucky
will get different-colored socks? Express your answer as a common fraction.
4. A bowl contains 10 jellybeans (four red, one blue, and five white). If you pick three
jellybeans from the bowl at random and without replacement, what is the
probability that exactly two will be red? Express your answer a common fraction.
5. Russell rolls a pair of dice with his eyes closed. His sister tells him that the sum of
dice is greater than or equal to nine. Knowing this information, Russell calculates
the probability that the he rolled doubles. What is that probability expressed as a
common fraction?
6. Each fair spinner below is divided into
four congruent regions. Joe used spinner
A. and Sally used spinner B. They added
the results. What is the probability that
the sum was even? Express your answer
as a common fraction.
1
2
5
7
3
Spinner A
4
16
9
Spinner B
7. On a canoe trip, the guide decides which two people will ride together in a canoe.
There are 12 people on the trip. Compute the probability that Kristen is assigned
the same canoe as her best friend Karen. Express your answer as a common
fraction.
8. Two positive integers are randomly chosen. Determine the probability that their
product is even. Express your answer as a common fraction.
9. Goran has a standard deck of 52 cards. He considers an ace to have a value of 1, a
jack has value of 11, a queen has a value of 12, a king has a value of 13, and all other
cards are their face value. What is the probability that a randomly selected card
will have an even value? Express your answer a common fraction.
10. The strips on the target shown are equal in width. If a
randomly-thrown dart lands on the target, what is the
probability that it will land within a black strip?
Express your answer as a common fraction.
11. A box contains six cards. Three of the cards are black on both sides, one card is
black on one side and red on the other, and two of cards are red on both sides. If
you pick a card at random from the box and see that the side facing you is red,
what is the probability that the other side is red? Express your answer as a
common fraction.
12. Students at Wooden High School go to school 180 days a year. Ms. Hines, the
geometry teacher, assigns homework 108 days a year. Mr. Chien, the biology
teacher, assigns homework 105 days a year. On a randomly selected day, what is the
probability that a student in Ms. Hine’s geometry class and in Mr. Chien’s biology
class will not have homework in either class? Express your answer as a common
fraction.
13. If you pick a letter at random from the alphabet, what is the probability that it is
in the word “mathematics”? Express your answer a common fraction.
14. A single ticket is randomly drawn from a set of tickets consecutively numbered 1
through 9999 inclusive. If you have tickets numbered from 5691 through 5699
inclusive, what is your probability of winning? Express your answer as a common
fraction.
15.
3
chance of making a free throw. What is the probability that she will
10
make both of her next two free throws? Express your answer as a common fraction.
Alex has a
16. The numerator of a fraction is randomly selected from the set {1, 3, 5, 7, 9}, and
the denominator is randomly selected from the set {1, 3, 5, 7, 9}. What is the
probability that the decimal representation of the resulting fraction is not a
terminating decimal? Express your answer as a common fraction.
17. A drawer contains eight red, eight yellow, eight green and eight black socks. What
is the probability of getting at least one pair of matching socks when five socks are
randomly pulled from the drawer?
18. A drawer contains 2 brown and 3 gray socks. The socks are taken out of the drawer
one at a time. What is the probability that the fourth sock removed is gray?
Express your answer as a common fraction.
19. If two distinct numbers are selected at random from the first seven prime
numbers, what is the probability that their sum is an even number? Express your
answer as a common fraction.
5
will be chosen at random. What is the
7
probability that the digit will be a 4? Express your answer a common fraction. (1998
Chapter Countdown).
20. One digit of the decimal representation of
21. A letter of the alphabet is randomly selected. What is the probability that the
letter chosen does not occur in the name of a month? Express your answer as a
common fraction.
22. Joe’s batting average is .323. What is the probability that he will get three hits in
three at-bats? Express your answer as a decimal to the nearest hundredth.
23. One-Fourth of Matilda’s candies are blue, 1/8 are green, 1/4 are yellow, and the
rest are red. Matilda selects one candy. What is the probability that the selected
candy is red? Express your answer as a common fraction.
Probability (Day 1 and 2) – Blue Solutions
1. The probability that a blue chip is drawn first and yellow chip second is 5/8 x 3/8 = 15/64. The
probability that the two chips selected are different colors is the sum of these two possibilities or
15/64 + 15/64 = 30/64 = 15/32.
2. The probability of drawing a red ball first, replacing it with a white ball first, and then drawing a red
2 1 1
x = . The probability of drawing a white ball, replacing it with a red ball, and then
4 4 8
2 3 3
x = . The combined probability that one of these two scenarios will occur
drawing a red ball is
4 4 8
1 3 1
is + = .
8 8 2
ball is
3. There are 3 choices for the first marble, and 2 choices for the second marble, so there are 6 ways
that the two marbles could have been drawn. Only one these ways shows blue first and yellow second.
The probability is 1/6.
4. The probability of getting two heads when two coins are flipped is one of four possible outcomes.
Those outcomes are HH, HT, TH, and TT. The probability is thus
5.
1
.
4
1
1024
6. There is only one way to get double 1’s. The probability of rolling a 1 on one dies is 1/6, so the
2
⎛1⎞
1
. That means that you would expect to get double 1’s only time
⎟ =
6
36
⎝ ⎠
probability of double 1’s is ⎜
out of 36 rolls. Since 144 = 4 • 36, we should expect to get double 1’s in 4 rolls.
7. We wish to know the probability that the product of two numbers rolled on dice have a product
greater than 8. A valid approach to probability problems with two independent events is to create a
chart to find the answer.
The chart to the right shows the results. Each die has
1 2 3 4 5 6
a possible outcome of an integer from 1 to 6, and those
possibilities are shown along the top and side of the
1 2 3 4 5 6
1
chart. Inside the chart are the products, and those
2 2 4 6 8 10 12
products that exceed 8 are italicized. Counting, there
are 20 italicized numbers, and there are 36 numbers
3 3 6 9 12 15 18
total in the chart, so the probability that a product
exceeds 8 is
20 5
= .
36 9
12 16 20 24
4
4
8
5
5
10 15 20 25 30
6
6
12 18 24 30 36
8. The following table shows the 24 possible sums of the dice. Six of these sums are less than five, so
the probability is
6
1
, or .
24
4
+
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
9. The teacher could choose any of the five volunteers first, followed by any of the remaining four
volunteers second. Once the two are chosen, however, we don’t actually care who was chosen first.
Therefore, we have to divide by the number of ways that the same two people could be chosen. This is
called a combination. We say “five choose two” and calculate as follows: (5 x 4) ÷ (2 X 1) = 20 ÷ 2 = 10
ways to choose two volunteers from the group of five. Only one of these pairs is the two girls, so the
probability is 1/10 that they will selected.
10. Rolling two standard 6-sided dice, there is 1 way to get a sum of 12, 2 ways to get a sum of 11, and 3
ways to get a sum of 10. That’s 6 out 36 possible outcomes for a probability of 1/6.
11. The least positive integer that is both a multiple of 7 and 11 is 77, which is larger than 70. Therefore,
the two events cannot happen together, so the probability is 0.
12. The smallest possible sum Tara can get when she rolls three dice is 3. This means that every one of
the 6 x 6 x 6 = 216 possible rolls will have a sum of three or more, and the probability is 100%.
13. There are eight outcomes when a coin is flipped three times. Six of those eight ways contain either
two heads or two tails. Hence, the probability is
6 3
= .
8 4
14. 0.36
Dependent Events
1. There are two sums that equal seven (3 + 4 and 2 + 5) and we don’t care about the order in which they
are picked. To get the 3 and the 4, we have a
and a
4 is
2
chance of getting one of the cards on the first pick
4
1
chance of getting the other card on the second pick. The probability of picking the 3 and the
3
2 1 1
1 1 1
x = . The same is true for the 2 and 5 pair. Combining these we get
+ = .
4 3 6
6 6 3
2. There are four of each our six primes ( 2, 3, 5, 7, J and K) in a deck of cards (one of each in each of
four suits) for a total of 4 x 6 = 24 prime cards. We would be happy to choose any one of them for our
first card. Similarly, only 16 good primes remain for our third card and only 12 remain for our fourth
card. The probability of drawing a Magic Prime Hand is thus
24 20 16 12
x
x
x
= 0.01418 or 0.014 to the
52 51 50 49
nearest thousandth.
3. Lucky could select a red sock first and a blue sock second or a blue sock first and a red sock second.
The probability of a red, then a blue is (5/13) x (8/12) = 40/156. The probability of a blue then a red
is (8/13) x (5/12) = 40/156. The probability that Lucky selects two different color socks is the sum of
these two separate probabilities or 20/39.
4. Let R stand for “red” and N stand for “not red”. If you pick three jellybeans from the bowl, there are
three ways that exactly two could be red, namely RRN, RNR and NRR. The probability of choosing RRN
is 4/10 x 3/9 x 6/8 = 72/720 = 1/10. The probability of choosing RNR is 4/10 x 6/9 x 3/8 = 72/720 =
1/10. The probability of choosing NRR is 6/10 x 4/9 x 3/8 = 72/720 = 1/10. Combining these different
ways of getting exactly two red jellybeans, we get an overall probability of 1/10 + 1/10 + 1/10 = 3/10.
5. There are four ways to roll a sum of 9 ; 6 + 3,5 + 4,4 +5 and 3 + 6. There are three ways to roll a sum
of 10 ; 6 + 4,5 + 5 and 4 + 6. There are two ways to roll a sum of 11 ; 6 + 5 and 5 + 6. And there is one
way to roll a sum of 12 ; 6 + 6. Of these 10 possible rolls that are greater than or equal to 9, only two
ways are doubles, so the probability that Russell rolled doubles is 2/10 or 1/5.
6. If the sum of the numbers is even, then either both of them spun an odd number or both of them spun
an even number. The probability of Joe spinning an even number is
spinning an even numbers
happening is
3
, and the probability of Sally
4
1
. Since these two events are independent, the probability of them both
2
3 1 3
x = . Similarly, the probability of Joe spinning an odd number is
4 2 8
probability of Sally spinning an odd is
1
, and the
4
1
1 1 1
; thus the probability of both happening is x = . The two
2
4 2 8
events – both spinning odds or both spinning evens – are mutually exclusive, since they cannot happen
simultaneously. Thus add the respective probabilities. The probability that the sum is even, then, is
3 1 4 1
+ = = .
8 8 8 2
7.
1
11
8.
3
4
9. There are six cards with an even value in each suit of 13 cards, so the probability that a randomly
selected card has an even value is 6/13.
10. Dividing the figure into a 6 x 6 grid with 36 small squares
allows you to account the shaded strips as unit squares. There
are 21 shaded squares, and 36 squares in the entire figure. The
probability that a randomly thrown dart lands within the
21 7
shaded region is
=
.
36 12
11. There are five different red sides that might be the one you are looking at. Four of those red sides
have red on the other side too. Only one of the red sides has black on the other side. The probability
is 4/5 that the other side is red, too.
12.
1
6
13.
4
13
14.
1
1111
15.
The probability that Alex makes both shots is
3 3
9
x =
.
10 10 100
16. Since both the numerator and denominator are randomly selected from the same set {1, 3, 5, 7, 9} and
the set has five elements, there are only 5 x 5 = 25 possible fractions. Fractions with terminating
decimal expressions have denominators with only powers of 2 or 5. The only possible denominator for a
reduced fraction in this set is 5. Thus, any of the five fractions with denominator 5 will have a
terminating decimal. In addition, any of the five fractions with denominator 1 is a whole number, which
also yields a terminating decimal. Finally, we must also look at fractions that could be reduced to give a
1
5
3 7 9
9
terminating decimal. These are and
, which have already been counted, and
,
,
, and
.
1
5
3 7 3
9
Thus, there are 5 + 5 + 4 = 14 fractions that terminate, so there are 25 – 14 = 11 fractions which do
not. Therefore, the probability that the decimal representation of the fraction is not a terminating
11
decimal is
.
25
17. The probability of getting at least one pair of matching socks is 1. Even if the first four socks chosen
are all different colors, the fifth sock will match one of them because there are only four colors.
18. There are 5C2 = 10 possible outcomes, and they can be listed easily, GGGBB, BGGGB, GGBGB, GBGGB,
BBGGG, BGGBG, BGBGG, GBGBG, GBBGG, GGBBG. Out of the ten outcomes, the fourth sock removed is
6 3
grey in six of them. The probability that the fourth sock removed is gray is
= .
10 5
19. The first 7 prime numbers are 2, 3, 5, 7, 11, 13 and 17. If the sum of two numbers is even, then both
numbers must be even or both must be odd. There is only one even number on the list, so choosing two
even numbers is not possible. Choosing any other numbers from the first six primes will work, so the
probability of an even total is 6/7 x 5/6 = 5/7.
20. The decimal representation of 5/7 is a repeating decimal with exactly six distinct digits in the portion
that repeats. (5/7 = .714285714285714285 … or .714285 . This means that there are six equallylikely digits to be chosen. There is a 1/6 chance that the digit chosen will be a 4.
21. Write all the letters of the alphabet, and then cross out the letters as they occur in the months in
order. You then are left with only the five letters K, Q, W, X and Z. thus, the probability that the
5
chosen letter is not a letter of the alphabet is
.
26
22. The probability that Joe will get a hit at any one of the three at bats is 0.323, so the probability he
will get a hit in each of three at-bats is (0.323)3 ≈ 0.03.
23. We know that ¼ + 1/8 + ¼ = 5/8 of the candies are not red. Consequently, 1 – 5/8 = 3/8 of the candies
are red. The probability of selecting a red candy is 3/8.
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some
cases, sources were not known.
Problems
all
Bibliography Information
Math Counts (http://mathcounts.org)