Grade 10 - Probability

(Sample Printed worksheet)

Answer The Following

1)Two dice are rolled. What is the probability that the two numbers add up to a prime number?
2)A bag contains 3 red balls, 5 blue balls, and 7 green balls. Carol draws 2 balls out of the bag. What is the probability that she gets a red ball and a blue ball?
3)Coin A is flipped 3 times and coin B is flipped 4 times. What is the probability that the number of heads obtained from flipping the two coins is the same?
4)A tyre manufacturer keeps the record of how much distance the tyres manufactured by the company travel before failing. They find the following data
Distance traveled in kilometersNumber of failing tyres
Less than 5000 80
5000 to 10000 300
10001 to 20000 360
20001 to 40000 504
More than 40000 756

If Sandra buys a tyre from them, what is the probability that it will last more than 20000 kilometers?
5)What is the probability that a leap year will contain 53 Tuesdays?

Choose correct answer(s) from given choice

6)A poll is taken among people working in Atlanta. The aim was to see what their annual salaries were.
Annual SalaryNumber of people
Less than 40000 360
40001 to 75000 820
75001 to 150000 1940
150001 to 250000 2752
More than 250000 4128

If you choose a person at random from this group, what is the probability that he or she earns more than 75000 annually?

a.  
8800
10000
 
b.  
8820
10000
 
c.  
8793
10000
 
d.  
8786
10000
 
7)Nancy selects 3 numbers randomly from the following set of 5 numbers
5, 3, 1, 6 and 9.
He puts them in the form of a proper fraction of the type  
ab
c
 .
What is the probability that you will get a fraction greater than  
517
18
 ?
a.  
12
30
 
b.  
6
30
 
c.  
6
36
 
d.  
18
36
 
8)Michelle is participating in a race. The probability that she will come first in the race is 0.3. The probability that she will come second in the race is 0.2. The probability that she will come in 3rd is 0.4, and the probability that she will be 4th is 0.1. What is the probability that she will win 2nd position or better in the race?
a. 0.5 b. 0.55
c. 0.45 d. 0.6
9)There are 4 pens belonging to 4 students. The pens were put into a box, and each student pulls out an pen one after the other. What is the probability that each student gets his or her own pen?
a.  
3
12
 
b.  
1
24
 
c.  
1
8
 
d.  
1
2
 
10)If a prime number is less than 23, what is the probability that it is also less than 17.
a. 7/8 b. 2/3
c. 3/4 d. 5/8

Answers

1)  
15
36
 
Step 1
The two dice that are rolled can show any of these values
Dice 1 : 1, 2, 3, 4, 5, 6
Dice 2 : 1, 2, 3, 4, 5, 6
So we can get a total of 36 combinations between them (6 x 6)
Step 2
If we take one value from the list of possible values from each Dice, we get numbers ranging from 2 (when both Dice show 1) to 12 (when both dice show 6).
Let's enumerate the prime numbers between 2 and 12. They are 2, 3, 5, 7 and 11
We need to see in how many ways we can get each of these values
Let's put the value rolled by the dice as (x,y), where x is the value rolled by Dice 1, and y the value rolled by Dice 2
- 2: The only way to get this is when we roll (1,1). 1 possibility
- 3: We can get this by (1,2) or (2,1). 2 possibilities.
- 5: We can get this by (2,3), (3,2), (1,4) or (4,1). 4 possibilities.
- 7: We can get this by (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). 6 possibilities.
- 11: We can get this by (5,6), or (6,5). 2 Possibilities
This gives us a total of 1 + 2 + 4 + 6 + 2 = 15 possible ways to get a prime number
Step 3
So the probability of getting the two numbers add up to a prime is  
15
36
 

2)  
15
105
 
Step 1
There are a total of 3+5+7 = 15 balls in the bag
Step 2
The number of ways to pick out 2 balls from a set of 15 balls is 15 C2 = 15 x (15-1)/2 = 105
Step 3
The number of ways to pick a red ball and a blue ball is obtained by multiplying the number of the balls of each of these colors. This is 3 x 5 = 15
Step 4
The probability that she gets a red ball and a blue ball is therefore  
15
105
 

3)  
35
128
 

4)  
1260
2000
 
Step 1
First we need to find the total number of tyres that are given here
Step 2
We add the number of tyres 80 + 300 + 360 + 504 + 756 = 2000
Step 3
To find the probability that the tyre Sandra purchased would last more than 20000 kilometers, we need to add the number of tyres that lasted more than 20000 kilometers
This is 504 + 756 = 1260
Step 4
The probability that the tyre lasts more than 20000 km =  
1260
2000
 

5)  
2
7
 
Step 1
There are 366 days in a leap year
Step 2
If we divide 366 by 7 (since there are seven days in a week), we will get an answer of 52, with a remainder of 2
This means that a leap year will have 52 Sundays, 52 Mondays, 52 Tuesdays, 52 Wednesdays, 52 Thursdays, 52 Fridays and 52 Saturdays.
Apart from these there will be two other days. This means that there will be two weekdays that occur 53 times.
Step 3
The two days could be (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday), (Friday, Saturday), or (Saturday, Sunday) - a total of seven combinations
Step 4
Out of these seven combinations, two of them have a Tuesday
Step 5
So the probability of either of those two days being a Tuesday is  
2
7
 

6) b.  
8820
10000
 
Step 1
First we need to find the total number of people among whom the poll was taken
Step 2
We add the number of people in the various sets 360 + 820 + 1940 + 2752 + 4128 = 10000
Step 3
To find the probability that the the random person chosen has a salary of more than 75000 , we need to add the number of people who have salaries greater than this number
This is 1940 + 2752 + 4128 = 8820
Step 4
So 8820 people out of a total of 10000 earn an annual salary greater than 75000
The probability that the randomly chosen person has a salary greater than 75000 =  
8820
10000
 

7) a.  
12
30
 
Step 1
We need to select 3 numbers out of the 5 given in order to get a fraction of the form  
ab
c
 
We are also told it is a proper fraction, so b should be greater than c
Step 2
Let's put the integers in a sorted manner.
We get 1, 3, 5, 6 and 9
Step 3
Now we need to see how many proper fractions can be formed from them
A proper fraction of the type  
ab
c
  has three integers, a whole number a, a numerator b, and a denominator c
Let's assume we use one of the integers in the list above as the whole number a
We now can select b and c from the remaining 4 integers
We can select 2 integers from 4 in 4C2 =  
4x3
2x1
  = 6 ways
For any pair we select, one will be greater than the other, and the smaller integer will form the numerator and the larger one the denominator
- Note: that the other way won't work - if the numerator is larger than the denominator it is not a proper fraction
So for each of the 5 integers, if we select one as the whole number, we get 6 possible combinations of numerator and denominator that can form a proper fraction
This means there are 5x6 = 30 possible proper fractions of the form  
ab
c
  that can be formed from these 5 integers
Step 4
Now we need to figure out how many of these 30 fractions are greater than  
517
18
 
In gt, we see that the whole number is 5, the numerator is 17 and the denominator is 18
We see that the numerator 17 is larger than the largest number in the set of numbers given to us, and the denominator 18 is one larger than 17
The implication of this is that  
517
18
  will be larger than any fraction that can be formed from the set of numbers 1, 3, 5, 6 and 9 where the whole number of the fraction is 5
So we only need to count the fractions that have the whole number greater than 5
These are the fractions that will have the whole numbers as 6 and 9
Step 5
Now, remember there are 30 proper fractions you can form from this list of number
From our analysis above, we also saw that for each number selected as the whole number, we can form 6 fractions from this list of numbers
Step 6
So using each of the numbers from 6 and 9 as the whole number, we can form 6 proper fractions
The total number of fractions that can be formed using 6 and 9 as the whole number = 6 x 2 = 12
Step 7
Out of the 30 fractions, 12 will be greater than  
517
18
 
Step 8
The probability is therefore =  
12
30
 

8) a. 0.5
Step 1
We are looking for the probability that she will come 2nd or better in the race
This is the probability that she will either win the race or be 2nd
Since she can be first or second, the probabilities can be added
Step 2
The probability that she will be first is 0.3
The probability that she will be in second place is 0.2
Step 3
Adding them, we get 0.3 + 0.2 = 0.5

9) b.  
1
24
 
Step 1
We have 4 pens and 4 students. Let's first see in how many different ways can the pens be distributed among the students
Step 2
We know that 4 objects can be distributed in 4 ! = 4 x 3 x...x 1 = 24 ways
Step 3
Out of these 24 ways, there is only one distribution where each student got his or her own pen
Step 4
So the probability of each student getting his or her own pen =  
1
24
 

10) c. 3/4
Step 1
There are 8 prime numbers (2, 3, 5, 7, 11, 13, 17, 19) which are less than 23
Of these 6 are less than 17
Step 2
Therefore probability is 6/8 = 3/4

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