# RD Sharma (2017) Solutions for Class 9 Math Chapter 25 - Probability

RD Sharma (2017) Solutions for Class 9 Math Chapter 25 Probability are provided here with simple step-by-step explanations. These solutions for Probability are extremely popular among class 9 students for Math Probability Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RD Sharma (2017) Book of class 9 Math Chapter 25 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RD Sharma (2017) Solutions. All RD Sharma (2017) Solutions for class 9 Math are prepared by experts and are 100% accurate.

#### Page No 25.13:

#### Question 1:

A coin is tossed 1000 times with the following frequencies:

Head: 455, Tail: 545

Compute the probability for each event.

#### Answer:

The coin is tossed 1000 times. So, the total number of trials is 1000.

Let *A *be the event of getting a head and *B *be the event of getting a tail.

The number of times *A* happens is 455 and the number of times *B* happens is 545.

Remember the empirical or experimental or observed frequency approach to probability.

If *n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have

#### Page No 25.13:

#### Question 2:

Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:

Two heads: 95 times

One tail: 290 times

No head: 115 times

Find the probability of occurrence of each of these events.

#### Answer:

The total number of trials is 500.

Let *A *be the event of getting two heads, *B* be the event of getting one tail and *C *be the event of getting no head.

The number of times *A* happens is 95, the number of times *B* happens is 290 and the number of times *C* happens is 115.

Remember the empirical or experimental or observed frequency approach to probability.

If *n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have

#### Page No 25.13:

#### Question 3:

Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes:

Outcome: | No head | One head | Two heads | Three heads |

Frequency: | 14 | 38 | 36 | 12 |

If the three coins are simultaneously tossed again, compute the probability of:

(i) 2 heads coming up.

(ii) 3 heads coming up.

(iii) at least one head coming up.

(iv) getting more heads than tails.

(v) getting more tails than heads.

#### Answer:

The total number of trials is 100.

Remember the empirical or experimental or observed frequency approach to probability.

If *n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A *be the event of getting two heads.

The number of times *A* happens is 36.

Therefore, we have

(ii) Let* B* be the event of getting three heads

The number of times *B* happens is 12.

Therefore, we have

(iii) Let *C* be the event of getting at least one head.

The number of times *C* happens is.

Therefore, we have

(iv) Let *D* be the event of getting more heads than tails.

The number of times *D* happens is.

Therefore, we have

(v) Let *E *be the event of getting more tails than heads.

The number of times *E* happens is.

Therefore, we have

#### Page No 25.13:

#### Question 4:

1500 families with 2 children were selected randomly and the following data were recorded:

Number of girls in a family | 0 | 1 | 2 |

Number of families | 211 | 814 | 475 |

If a family is chosen at random, compute the probability that it has:

(i) No girl

(ii) 1 girl

(iii) 2 girls

(iv) at most one girl

(v) more girls than boys

#### Answer:

The total number of trials is 1500.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A *be the event of having no girl.

The number of times *A* happens is 211.

Therefore, we have

(ii) Let* B* be the event of having one girl.

The number of times *B* happens is 814.

Therefore, we have

(iii) Let *C* be the event of having two girls.

The number of times *C* happens is 475.

Therefore, we have

(iv) Let *D* be the event of having at most one girl.

The number of times *D* happens is.

Therefore, we have

(v) Let *E *be the event of having more girls than boys.

The number of times *E* happens is 475.

Therefore, we have

#### Page No 25.13:

#### Question 5:

In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays.

(i) he hits boundary

(ii) he does not hit a boundary.

#### Answer:

The total number of trials is 30.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A *be the event of hitting boundary.

The number of times *A* happens is 6.

Therefore, we have

(ii) Let* B* be the event of does not hitting boundary.

The number of times *B* happens is.

Therefore, we have

#### Page No 25.13:

#### Question 6:

The percentage of marks obtained by a student in monthly unit tests are given below:

Unit test: | I | II | III | IV | V |

Percentage of marks obtained: | 69 | 71 | 73 | 68 | 76 |

Find the probability that the student gets:

(i) more than 70% marks

(ii) less than 70% marks

(iii) a distinction

#### Answer:

The total number of trials is 5.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A *be the event of getting more than 70% marks.

The number of times *A* happens is 3.

Therefore, we have

(ii) Let* B* be the event of getting less than 70% marks.

The number of times *B* happens is 2.

Therefore, we have

(iii) Let *C* be the event of getting a distinction.

The number of times *C* happens is 1.

Therefore, we have

#### Page No 25.14:

#### Question 7:

To know the opinion of the students about Mathematics, a survey of 200 students was conducted. The data is recorded in the following table:

Opinion: | Like | Dislike |

Number of students: | 135 | 65 |

Find the probability that a student chosen at random (i) likes Mathematics (ii) does not like it.

#### Answer:

The total number of trials is 200.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A *be the event of liking mathematics.

The number of times *A* happens is 135.

Therefore, we have

(ii) Let* B* be the event of disliking mathematics.

The number of times *B* happens is 65.

Therefore, we have

#### Page No 25.14:

#### Question 8:

The blood groups of 30 students of class IX are recorded as follows:

A | B | O | O | AB | O | A | O | B | A | O | B | A | O | O |

A | AB | O | A | A | O | O | AB | B | A | O | B | A | B | O |

A student is selected at random from the class from blood donation, Fin the probability that the blood group of the student chosen is:

(i) A

(ii) B

(iii) AB

(iv) O

#### Answer:

The total number of trials is 30.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A*_{1}* *be the event that the blood group of a chosen student is A.

The number of times *A*_{1} happens is 9.

Therefore, we have

(iii) Let* **A*_{2} be the event that the blood group of a chosen student is B.

The number of times *A*_{2} happens is 6.

Therefore, we have

(iii) Let *A*_{3}* *be the event that the blood group of a chosen student is AB.

The number of times *A*_{3} happens is 3.

Therefore, we have

(iv) Let* **A*_{4} be the event that the blood group of a chosen student is O.

The number of times *A*_{4} happens is 12.

Therefore, we have

#### Page No 25.14:

#### Question 9:

Eleven bags of wheat flour, each marked 5 Kg, actually contained the following weights of flour (in kg):

4.97, 5.05, 5.08, 5.03, 5.00, 5.06, 5.08, 4.98, 5.04, 5.07, 5.00

Find the probability that any of these bags chosen at random contains more than 5 kg of flour.

#### Answer:

The total number of trials is 11.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Let *A*_{1}* *be the event that the actual weight of a chosen bag contain more than 5 Kg of flour.

The number of times *A*_{1} happens is 7.

Therefore, we have

#### Page No 25.14:

#### Question 10:

Following table shows the birth month of 40 students of class IX.

Jan. | Feb | March | April | May | June | July | Aug. | Sept. | Oct. | Nov. | Dec. |

3 | 4 | 2 | 2 | 5 | 1 | 2 | 5 | 3 | 4 | 4 | 4 |

Find the probability that a student was born in August.

#### Answer:

The total number of trials is 40.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Let *A*_{1}* *be the event that the birth month of a chosen student is august.

The number of times *A*_{1} happens is 5.

Therefore, we have

#### Page No 25.14:

#### Question 11:

Given below is the frequency distribution table regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days.

Conc. of SO_{2} |
0.00-0.04 | 0.04-0.08 | 0.08-0.12 | 0.12-0.16 | 0.16-0.20 | 0.20-0.24 |

No. days: | 4 | 8 | 9 | 2 | 4 | 3 |

Find the probability of concentration of sulphur dioxide in the interval

0.12-0.16 on any of these days.

#### Answer:

The total number of trials is 30.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Let *A*_{1}* *be the event that the concentration of sulphur dioxide in a day is 0.12-0.16 parts per million.

The number of times *A*_{1} happens is 2.

Therefore, we have

#### Page No 25.14:

#### Question 12:

A company selected 2400 families at random and survey them to determine a relationship between income level and the number of vehicles in a home. The information gathered is listed in the table below:

Monthly income:(in Rs) |
Vehicles per family |
|||

0 | 1 | 2 | Above 2 | |

Less than 7000 7000-10000 10000-13000 13000-16000 16000 or more |
10 0 1 2 1 |
160 305 535 469 579 |
25 27 29 29 82 |
0 2 1 25 88 |

If a family is chosen, find the probability that family is:

(i) earning Rs10000-13000 per month and owning exactly 2 vehicles.

(ii) earning Rs 16000 or more per month and owning exactly 1 vehicle.

(iii) earning less than Rs 7000 per month and does not own any vehicle.

(iv) earning Rs 13000-16000 per month and owning more than 2 vehicle.

(v) owning not more than 1 vehicle

(vi) owning at least one vehicle.

#### Answer:

The total number of trials is 2400.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A*_{1}* *be the event that a chosen family earns Rs 10000-13000 per month and owns exactly 2 vehicles.

The number of times *A*_{1} happens is 29.

Therefore, we have

(ii) Let *A*_{2}* *be the event that a chosen family earns Rs 16000 or more per month and owns exactly 1 vehicle.

The number of times *A*_{2} happens is 579.

Therefore, we have

(iii) Let *A*_{3}* *be the event that a chosen family earns less than Rs 7000 per month and does not owns any vehicles.

The number of times *A*_{3} happens is 10.

Therefore, we have

(iv) Let *A*_{4}* *be the event that a chosen family earns Rs 13000-16000 per month and owns more than 2 vehicles.

The number of times *A*_{4} happens is 25.

Therefore, we have

(v) Let *A*_{5}* *be the event that a chosen family owns not more than 1 vehicle (may be 0 or 1). In this case the number of vehicles is independent of the income of the family.

The number of times *A*_{5} happens is

.

Therefore, we have

(vi) Let *A*_{6}* *be the event that a chosen family owns atleast 1 vehicle (may be 1 or 2 or above 2). In this case the number of vehicles is independent of the income of the family.

The number of times *A*_{6} happens is

.

Therefore, we have

#### Page No 25.15:

#### Question 13:

The following table gives the life time of 400 neon lamps:

Life time (in hours) |
300-400 | 400-500 | 500-600 | 600-700 | 700-800 | 800-900 | 900-1000 |

Number of lamps: | 14 | 56 | 60 | 86 | 74 | 62 | 48 |

A bulb is selected of random, Find the probability that the the life time of the selected bulb is:

(i) less than 400

(ii) between 300 to 800 hours

(iii) at least 700 hours.

#### Answer:

The total number of trials is 400.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A*_{1}* *be the event that the lifetime of a chosen bulb is less than 400 hours.

The number of times *A*_{1} happens is 14.

Therefore, we have

(ii) Let *A*_{2}* *be the event that the lifetime of a chosen bulb is in between 300 to 800 hours.

The number of times *A*_{2} happens is.

Therefore, we have

.

(iii) Let *A*_{3}* *be the event that the lifetime of a chosen bulb is atleast 700 hours.

The number of times *A*_{3} happens is.

Therefore, we have

#### Page No 25.15:

#### Question 14:

Given below is the frequency distribution of wages (in Rs) of 30 workers in a certain factory:

Wages (in Rs) | 110-130 | 130-150 | 150-170 | 170-190 | 190-210 | 210-230 | 230-250 |

No. of workers | 3 | 4 | 5 | 6 | 5 | 4 | 3 |

A worker is selected at random. Find the probability that his wages are:

(i) less than Rs 150

(ii) at least Rs 210

(iii) more than or equal to 150 but less than Rs 210.

#### Answer:

The total number of trials is 30.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

(i) Let *A*_{1}* *be the event that the wages of a worker is less than Rs 150.

The number of times *A*_{1} happens is.

Therefore, we have.

(ii) Let *A*_{2}* *be the event that the wages of a worker is atleast Rs 210.

The number of times *A*_{2} happens is.

Therefore, we have.

(iii) Let *A*_{3}* *be the event that the wages of a worker is more than or equal to Rs 150 but less than Rs 210.

The number of times *A*_{3} happens is.

Therefore, we have

#### Page No 25.16:

#### Question 1:

*Mark the correct alternative in each of the following:*

The probability of an impossible event is

(a) 1

(b) 0

(c) less than 0

(d) greater than 1

#### Answer:

We have to find the probability of an impossible event.

Note that the number of occurrence of an impossible event is 0. This is the reason that’s why it is called impossible event.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Note that *n *is a positive integer, it can’t be zero. So, whatever may be the value of *n*, the probability of an impossible event is.

Hence the correct option is **(b)**.

#### Page No 25.16:

#### Question 2:

The probability of a certain event is

(a) 0

(b) 1

(c) greater than 1

(d) less than 0

#### Answer:

We have to find the probability of a certain event.

Note that the number of occurrence of an impossible event is same as the total number of trials. When we repeat the experiment, every times it occurs. This is the reason that’s why it is called certain event.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Note that *n *is a positive integer, it can’t be zero. So, the probability of an impossible event is.

Hence the correct option is **(b)**.

#### Page No 25.16:

#### Question 3:

The probability an event of a trial is

(a) 1

(b) 0

(c) less than 1

(d) more than 1

#### Answer:

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Note that *m *is always less than or equal to *n *and *n* is a positive integers, it can’t be zero. But, *m *is a non negative integer. So, the maximum value of probability of an event is, which is the probability of a certain event and the minimum value of it is 0, which is the probability of an impossible event. For any other events the value is in between 0 and 1.

Hence the correct option is **(c)**.

#### Page No 25.16:

#### Question 4:

Which of the following cannot be the probability of an event?

(a) $\frac{1}{3}$

(b) $\frac{3}{5}$

(c) $\frac{5}{3}$

(d) 1

#### Answer:

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Note that *m *is always less than or equal to *n *and *n* is a positive integers, it can’t be zero. But, *m *is a non negative integer. So, the maximum value of probability of an event is, which is the probability of a certain event and the minimum value of it is 0, which is the probability of an impossible event. For any other events the value is in between 0 and 1.

All the options except (c) satisfy the above criteria’s.

Hence the correct option is **(c)**.

#### Page No 25.16:

#### Question 5:

Two coins are tossed simultaneously. The probability of getting atmost one head is

(a) $\frac{1}{4}$

(b) $\frac{3}{4}$

(c) $\frac{1}{2}$

(d) $\frac{1}{4}$

#### Answer:

The random experiment is tossing two coins simultaneously.

All the possible outcomes are HH, HT, TH, and TT.

Let *A *be the event of getting at most one head.

The number of times *A *happens is 3.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have

So, the correct choice is **(b)**.

#### Page No 25.16:

#### Question 6:

A die is thrown 100 times. If the probability of getting an even number is $\frac{2}{5}$. How many times an odd number is obtained?

#### Answer:

The total number of trials is 100. Let the number of times an even number is obtained is *x*.

Let *A *be the event of getting an even number.

The number of times *A* happens is *x*.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have.

But, it is given that. So, we have

Hence an even number is obtained 40 times. Consequently, an odd number is obtainedtimes.

#### Page No 25.16:

#### Question 7:

Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes:

Outcome |
3 heads | 2 heads | 1 head | No head |

Frequency |
23 | 72 | 77 | 28 |

Find the probability of getting at most two heads.

#### Answer:

The total number of trials is 200.

Let *A *be the event of getting at most two heads.

The number of times *A* happens is.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have.

#### Page No 25.16:

#### Question 8:

In Q.No. 7, what is the probability of getting at least two heads?

#### Answer:

The total number of trials is 200.

Let *A *be the event of getting atleast two heads.

The number of times *A* happens is.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have

#### Page No 25.17:

#### Question 6:

A coin is tossed 1000 times, if the probability of getting a tail is 3/8, how many times head is obtained?

(a) 525

(b) 375

(c) 625

(d) 725

#### Answer:

The total number of trials is 1000. Let *x *be the number of times a tail occurs.

Let *A *be the event of getting a tail.

The number of times *A* happens is *x*.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have.

But, it is given that. So, we have

Hence a tail is obtained 375 times.

Consequently, a head is obtainedtimes.

So, the correct choice is **(c)**.

#### Page No 25.17:

#### Question 7:

A dice is rolled 600 times and the occurrence of the outcomes 1, 2, 3, 4, 5 and 6 are given below:

Outcome |
1 | 2 | 3 | 4 | 5 | 6 |

Frequency |
200 | 30 | 120 | 100 | 50 | 100 |

The probability of getting a prime number is

(a) $\frac{1}{3}$

(b) $\frac{2}{3}$

(c) $\frac{49}{60}$

(d) $\frac{39}{125}$

#### Answer:

The total number of trials is 600.

Let *A *be the event of getting a prime number (2, 3 and5).

The number of times *A* happens is.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have

So, the correct choice is **(a)**.

#### Page No 25.17:

#### Question 8:

The percentage of attendance of different classes in a year in a school is given below:

Class: |
X | IX | VIII | VII | VI | V |

Attendance: |
30 | 62 | 85 | 92 | 76 | 55 |

What is the probability that the class attendance is more than 75%?

(a) $\frac{1}{6}$

(b) $\frac{1}{3}$

(c) $\frac{5}{6}$

(d) $\frac{1}{2}$

#### Answer:

The total number of trials is 6.

Let *A *be the event that the attendance of a class is more than 75%.

The number of times *A* happens is 3 (for classes’ VIII, VII and VI).

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have

So, the correct choice is **(d)**.

#### Page No 25.17:

#### Question 9:

A bag contains 50 coins and each coin is marked from 51 to 100. One coin is picked at random. The probability that the number on the coin is not a prime number, is

(a) $\frac{1}{5}$

(b) $\frac{3}{5}$

(c) $\frac{2}{5}$

(d) $\frac{4}{5}$

#### Answer:

The total number of trials is 50.

Let *A *be the event that the number on the picked coin is not a prime.

The prime’s lies in between 51 and 100 are 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97. They are 10 in numbers. Therefore the numbers lies between 51 and 100 and which are not primes are in numbers.

So, the number of times *A* happens is 40.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have

So, the correct choice is **(d)**.

#### Page No 25.17:

#### Question 10:

In a football match, Ronaldo makes 4 goals from 10 penalty kicks. The probability of converting a penalty kick into a goal by Ronaldo, is

(a) $\frac{1}{4}$

(b) $\frac{1}{6}$

(c) $\frac{1}{3}$

(d) $\frac{2}{5}$

#### Answer:

The total number of trials is 10.

Let *A *be the event that Ronaldo makes a goal in a penalty kick.

The number of times *A* happens is 4.

Remember the empirical or experimental or observed frequency approach to probability.

*n* be the total number of trials of an experiment and *A* is an event associated to it such that *A* happens in *m*-trials. Then the empirical probability of happening of event *A *is denoted byand is given by

Therefore, we have

So, the correct choice is **(d)**.

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