## Probability

### Class-10-Mathematics-1-Chapter-5-Maharashtra Board

### Solutions

**Practice Set 5.1**

**Question 1.1. **

**How many possibilities are there in each of the following?**

**(1) Vanita knows the following sites in Maharashtra. She is planning to visit one of them in her summer vacation.**

**Ajintha, Mahabaleshwar, Lonar Sarovar, Tadoba wild life sanctuary, Amboli, Raigad, Matheran, Anandavan.**

There are 8 places in list for planning to visit.

Therefore, Vanita has 8 possibilities of selecting any one of them.

**(2) Any day of a week is to be selected randomly.**

There are 7 days in a week.

Therefore, there are 7 possbilities of selecting any one day randomly.

**(3) Select one card from the pack of 52 cards.**

There are 52 cards.

Therefore, there are 52 possibilities of selecting one card.

**(4) One number from 10 to 20 is written on each card. Select one card randomly.**

There are 11 numbers from 10 to 20 written on the cards.

Therefore, there are 11 possibilities of selecting one card randomly.

**Practice Set 5.2**

**Question 2.1**

**For each of the following experiments write sample space ‘S’ and number of sample points n(S).**

**(1) One coin and one die are thrown simultaneously.**

**(2) Two digit numbers are formed using digits 2, 3 and 5 without repeating a digits.**

** (1)** The sample space for a coin

S = {H,T}, ∴ n(S)=2

The sample space for a die

S = (1, 2, 3, 4, 5, 6) ∴ n(S) = 6

∴ if a coin is tossed and a die is thrown simultaneously, the sample space

S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} ∴ **n(S) = 12**

**(2)** The two-digit numbers formed using digits 2,3,5

S = {23, 25, 32, 35, 52, 53}, ∴ **n(S) = 6.**

**Question 2.2.**

**The arrow is rotated and it stops randomly on the disc. Find out on which colour it may stop.**

There are six colours on the disc. The arrow can stop on any of the colours.

∴ S = {Red, Orange, Yellow, Blue, Green, Purple }

∴** n(S)= 6.**

** ****Question 2.3.**

**In the month of March 2019, find the days on which the date is a multiple of 5. (see the given page of the calendar)**

The dates multiples of 5 are 5, 10, 15, 20, 25, 30.

S={Tuesday, Sunday, Friday, Wednesday, Monday, Saturday }

∴ n(S) = 6.

**Question 2.**4.

**Form a ‘Road safety commitee’ of two, from 2 boys (B1, B2) and 2 girls(G1, G2).**

**Complete the following activity to write the sample space.**

**(a) Committee of 2 boys = […..] **

**(b) Committee of 2 girls = [….]**

**(c) Committee of one boy and one girl = [B1G1] [….] […..] [……] **

**∴** **Sample space = {..., ..., ..., ..., ..., ...}**

(a) Committee of 2 boys = [B_{1 }B_{2}]

(b) Committee of 2 girls = [G_{1 }G_{2}]

(c) Committee of one boy and one girl = [B_{1 }G_{1}] [B_{1 }G_{2}] [B_{2 }G_{1}] [B_{2 }G_{2}]

∴ Sample space = { B_{1 }B_{2}, G_{1 }G_{2}, B_{1 }G_{1}, B_{1 }G_{2,} B_{2 }G_{1}, B_{2 }G_{2}}

**Practice Set 5.3**

**Question 3.1. **

**Write sample space ‘S’ and number of sample point n(S) for each of the following experiments. Also write events A, B, C in the set form and write n(A), n(B), n(C).**

**(1) One die is rolled,**

** Event A : Even number on the upper face.**

** Event B : Odd number on the upper face.**

** Event C : Prime number on the upper face.**

One die is rolled.

∴ the sample space S = {1, 2, 3, 4, 5, 6}. ∴ n(S) = 6.

Event A : Even number on the upper face.

∴ A = {2, 4, 6} ∴ n(A) = 3.

Event B : Odd number on the upper face.

∴ B = {1, 3, 5}. ∴ n(B) = 3

Event C : Prime number on the upper face.

∴ C = {2, 3, 5}, ∴ n(C) = 3.

**(2) Two dice are rolled simultaneously,**

** Event A : The sum of the digits on upper faces is a multiple of 6.**

** Event B : The sum of the digits on the upper faces is minimum 10.**

** Event C : The same digit on both the upper faces.**

S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

∴ n(S) = 36.

Event A : The sum of the digits on upper faces is a multiple of 6.

∴ A = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 6) }. ∴ n(A) =6.

Event B : The sum of the digits on upper faces is minimum 10.

∴ B = {(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6) }. ∴ n(B) = 6.

Event C : The same digit on both the upper faces.

∴ C= {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) }. ∴ n(C) = 6.

**(3) Three coins are tossed simultaneously.**

** Condition for event A : To get at least two heads.**

** Condition for event B : To get no head.**

** Condition for event C : To get head on the second coin.**

Three coins are tossed simultaneously.

∴ the sample space,

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. ∴ n(S) = 8.

Condition for event A : To get at least two heads.

∴ A = {HHH, HHT, HTH, THH} ∴ n(A) = 4.

Condition for event B : To get no head.

∴ B = {TTT} ∴ n(B) = 1.

Condition for event C: To get head on the second coin.

∴ C = {HHH, HHT, THH, THT } ∴ n(C) = 4.

**(4) Two digit numbers are formed using digits 0, 1, 2, 3, 4, 5 without repetition of the digits.**

** Condition for event A : The number formed is even**

** Condition for event B : The number formed is divisible by 3.**

** Condition for event C : The number formed is greater than 50.**

As we have to form two-digit numbers, 0 cannot be at tens place.

The sample space

S = {10, 12, 13, 14, 15, 20, 21, 23, 24, 25, 30, 31, 32, 34, 35, 40, 41, 42, 43, 45, 50, 51, 52, 53, 54}, ∴ n(S) = 25.

(i) Condition for event A : The number formed is even.

∴ A = {10, 12, 14, 20, 24, 30, 32, 34, 40, 42, 50, 52, 54}, **∴**** n(A) = 13**.

(ii) Condition for event B : The number is divisible by 3.

∴ B = {12, 15, 21, 24, 30, 42, 45, 51, 54}, ∴ **n(B) = 9**.

(iii) Condition for event C: The number is greater than 50.

∴ C = {51, 52, 53, 54}, ∴ **n(C) = 4**.

**(5) From three men and two women, environment committee of two persons is to be formed.**

**Condition for event A : There must be at least one woman member.**

**Condition for event B : One man, one woman committee to be formed.**

**Condition for event C : There should not be a woman member.**

Here, there are 3 men M_{1}, M_{2}, M_{3} and 2 women W_{1}, W_{2}.

A committee of two is to be formed.

∴ the sample space

S = {M_{1}M_{2}, M_{1}M_{3}, M_{2}M_{3}, M_{1}W_{1}, M_{1}W_{2}, M_{2}W_{1}, M_{2}W_{2}, M_{3}W_{1}, M_{3}W_{2}, W_{1}W_{2}}

∴ n(S) = 10

(i) Condition for event A : There must be at least one woman.

∴ A = {M_{1}W_{1}, M_{1}W_{2}, M_{2}W_{1}, M_{2}W_{2}, M_{3}W_{1}, M_{3}W_{2}, W_{1}W_{2}}, ∴ n(A) = 7.

(ii) Condition for event B : One man, one woman committee to be formed.

∴ B = {M_{1}W_{1}, M_{1}W_{2}, M_{2}W_{1}, M_{2}W_{2}, M_{3}W_{1}, M_{3}W_{2}}, ∴ n(B) = 6.

(iii) Condition for event C: There is no woman in the committee.

∴ C = {M_{1}M_{2}, M_{1}M_{3}, M_{2}M_{3}}, ∴ n(C) = 3.

**(6) One coin and one die are thrown simultaneously.**

**Condition for event A : To get head and an odd number.**

**Condition for event B : To get a head or tail and an even number.**

**Condition for event C : Number on the upper face is greater than 7 and tail on the coin.**

The sample space for a coin

S = {H, T}, ∴ n(S) = 2.

The sample space for a die

S = (1, 2, 3, 4, 5, 6), ∴ n(S) = 6

∴ if a coin is tossed and a die is thrown simultaneously, the sample space

S = {H1, H2, H3, H4, H5, H6,T1,T2,T3,T4,T5,T6}, ∴ n(S) = 12

(i) Condition for event A : To get a head and an odd number.

∴ A = {H1, H3, H5}, ∴ n(A) = 3.

(ii) Condition for event B : To get a head or tail and an even number.

∴ B = {H2, H4, H6, T2, T4, T6}. ∴ n(B) =6.

(iii) Condition for event C: Number on the upper face is greater than 7 and tail on the coin.

There is no number greater than 6 on a die.

∴ C = Φ, ∴ n(C) = 0.

**Practice Set 5.4**

**Question 4.1.**

**If two coins are tossed, find the probability of the following events.**

**(1) Getting at least one head. (2) Getting no head.**

Let S be the sample space.

Then S = {HH, HT, TH, TT}. ∴ n(S) = 4.

(1) Let A be the event where at least one head turns up

Then A = {HH, HT, TH}, ∴ n(A) = 3.

P(A) = \(\frac{n(A)}{n(S)}\) , ∴ P(A) = \(\frac{3}{4}\).

(2) Let B be the event where no head turns up.

Then B = {TT} , ∴ n(B) = 1

P(B) = \(\frac{n(B)}{n(S)}\) , ∴ P(B) = \(\frac{1}{4}\).

**Answer : (1) \(\frac{3}{4}\)** ** (2) \(\frac{1}{4}\)** **.**

**Question 4.2.**

** If two dice are rolled simultaneously, find the probability of the following events.**

**(1) The sum of the digits on the upper faces is at least 10.**

**(2) The sum of the digits on the upper faces is 33.**

**(3) The digit on the first die is greater than the digit on second die.**

Two dice are rolled simultaneously.

∴ the sample space is

S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

∴ n(S) = 36.

(1) Let A be the event that the sum of the digits on the upper faces is at least 10.

Then A = { (4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6) }, ∴ n(A) = 6.

∴ P(A) = \(\frac{n(A)}{n(S)} = \frac{6}{36}\) = \(\frac{1}{6}\).

(2) Let B be the event that the sum of the digits on the upper faces is 33.

The sum of the digits on the upper faces is maximum (6, 6) that is 12.

∴ B = { }. ∴ n(B) = 0

∴ P(B) = \(\frac{n(B)}{n(S)} = \frac{0}{36}\) = 0

(3) Let C be the event that the digit on the first die is greater than the digit on the second die.

Then C = { (2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5) }, ∴ n(C) = 15.

∴ P(C) = \(\frac{n(C)}{n(S)} = \frac{15}{36}\) = \(\frac{5}{12}\).

**Answer : (1) \(\frac{1}{6}\)** ** (2) 0 (3) \(\frac{5}{12}\)****.**

**Question 4.3.**

**There are 15 tickets in a box, each bearing one of the numbers from 1 to 15. One ticket is drawn at random from the box. Find the probability of event that the ticket drawn -**

**(1) shows an even number. (2) shows a number which is a multiple of 5.**

The sample space is

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, ∴ n(S) = 15

(1) Let A be the event that the ticket drawn bears an even number.

Then A = {2, 4, 6, 8, 10, 12, 14}, ∴ n(A) = 7

P(A) = \(\frac{n(A)}{n(S)} = \frac{7}{15}\)

(2) Let B be the event that the ticket drawn bears a number which is a multiple of 5.

Then B = {5, 10, 15}, ∴ n(B) = 3

P(B) = \(\frac{n(B)}{n(S)} = \frac{3}{15}\) = \(\frac{1}{5}\).

**Answer : (1) \(\frac{7}{15}\)** ** (2) \(\frac{1}{5}\)**

**Question 4.4. **

**A two digit number is formed with digits 2, 3, 5, 7, 9 without repetition. What is the probability that the number formed is**

**(1) an odd number ? (2) a multiple of 5 ?**

The sample space is

S= {23, 25, 27, 29, 32, 35, 37, 39, 52, 53, 57, 59, 72, 73, 75, 79, 92, 93, 95, 97}

∴ n(S) = 20

(1) Let A be the event that two-digit odd numbers are formed.

Then A = {23, 25, 27, 29, 35, 37, 39, 53, 57, 59, 73, 75, 79, 93, 95, 97},

∴ n(A) = 16

P(A) = \(\frac{n(A)}{n(S)} = \frac{16}{20}\) = \(\frac{4}{5}\).

(2) Let B be the event that the two-digit number is a multiple of five.

Then B = {25, 35, 75, 95}, ∴ n(B) = 4

P(B) = \(\frac{n(B)}{n(S)} = \frac{4}{20}\) = \(\frac{1}{5}\).

**Answer : (1) \(\frac{4}{5}\).** ** (2) \(\frac{1}{5}\).**

**Question 4.5.**

**A card is drawn at random from a pack of well shuffled 52 playing cards. Find the probability that the card drawn is -**

**(1) an ace. (2) a spade.**

S is the sample space

∴ n(S) = 52 ….... (There are 52 playing cards)

(1) Let A be the event that the card drawn is an ace.

There are four suits, Spade, Heart, Diamond and Club. Each suit has one ace.

(S) = { A-S, A-H, A-D, A-C}, ∴ n(A) = 4

P(A) = \(\frac{n(A)}{n(S)} = \frac{4}{52}\) = \(\frac{1}{13}\).

(2) Let B be the event that the card drawn is a spade.

There are 13 cards in the suit of spade One spade can be drawn out of 13 spade cards in 13 ways.

∴ n(B) = 13

P(B) = \(\frac{n(B)}{n(S)} = \frac{13}{52}\) = \(\frac{1}{4}\).

**Answer : (1) \(\frac{1}{13}\).** ** (2) \(\frac{1}{4}\).**

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