A coin is tossed 1000 times with the following frequencies:

Head: 455, Tail: 545

Compute the probability for each event.

**Answer
1** :

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 by and is given by

Therefore, we have

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
2** :

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 by and is given by

Therefore, we have

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
3** :

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 by and 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

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
4** :

The total number of trials is 1500.

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

(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

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
5** :

The total number of trials is 30.

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

(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

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
6** :

The total number of trials is 5.

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

(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

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
7** :

The total number of trials is 200.

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

(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

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
8** :

The total number of trials is 30.

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

(i) Let A1 be the event that the blood group of a chosen student is A.

The number of times A1 happens is 9.

Therefore, we have

(iii) Let A2 be the event that the blood group of a chosen student is B.

The number of times A2 happens is 6.

Therefore, we have

(iii) Let A3 be the event that the blood group of a chosen student is AB.

The number of times A3 happens is 3.

Therefore, we have

(iv) Let A4 be the event that the blood group of a chosen student is O.

The number of times A4 happens is 12.

Therefore, we have

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
9** :

The total number of trials is 11.

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

Let A1 be the event that the actual weight of a chosen bag contain more than 5 Kg of flour.

The number of times A1 happens is 7.

Therefore, we have

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
10** :

The total number of trials is 40.

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

Let A1 be the event that the birth month of a chosen student is august.

The number of times A1 happens is 5.

Therefore, we have

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