## NCERT Solutions for Class 11 Maths Chapter 16 Probability (Ex 16.1) Exercise 16.1

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## Access NCERT Solutions for Class 11 Maths Chapter 16- Probability

### Exercise 16.1

1. Describe the sample space for the indicated experiment: A coin is tossed three times.

Ans: A coin consists of two sides : head $\left( H \right)$ and tail $\left( T \right)$.

When a coin is tossed one time, the total number of outcomes are 2.

When a coin is tossed three times then the total number of outcomes would be $2 \times 2 \times 2$.

Thus the total number of outcomes when a coin is tossed three times is 8.

Therefore the sample space are written as:

$S = \left\{ {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} \right\}$

2. Describe the sample space for the indicated experiment: A dice is thrown two times.

Ans: The possible outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

When a dice is thrown one time, the total number of outcomes are 6.

When a dice is thrown two times then the total number of outcomes would be $6 \times 6$.

Thus the total number of outcomes when dice is thrown two times is 36.

Therefore the sample space are written as:

\[S = \{ \left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right),\left( {1,4} \right),\left( {1,5} \right),\left( {1,6} \right),\left( {2,1} \right),\left( {2,2} \right), \]

$ \left( {2,3} \right),\left( {2,4} \right),\left( {2,5} \right),\left( {2,6} \right),\left( {3,1} \right),\left( {3,2} \right),\left( {3,3} \right),\left( {3,4} \right), $

$ \left( {3,5} \right),\left( {3,6} \right),\left( {4,1} \right),\left( {4,2} \right),\left( {4,3} \right),\left( {4,4} \right),\left( {4,5} \right),\left( {4,6} \right), $

$ \left( {5,1} \right),\left( {5,2} \right),\left( {5,3} \right),\left( {5,4} \right),\left( {5,5} \right),\left( {5,6} \right),\left( {6,1} \right),\left( {6,2} \right), $

$ \left( {6,3} \right),\left( {6,4} \right),\left( {6,5} \right),\left( {6,6} \right) \}$

3. Describe the sample space for the indicated experiment: A coin is tossed four times.

Ans: A coin consists of two sides : head $\left( H \right)$ and tail $\left( T \right)$.

When a coin is tossed one time, the total number of outcomes are 2.

When a coin is tossed four times then the total number of outcomes would be $2 \times 2 \times 2 \times 2$.

Thus the total number of outcomes when a coin is tossed three times is 16.

Therefore the sample space are written as:

$S =\{ HHHH,HHHT,HHTH,HHTT,HTHH,HTHT,HTTH,TTHH $

$ HTTT,THHH,THHT,THTH,THTT,TTHT,TTTH,TTTT \}$

4. Describe the sample space for the indicated experiment: A coin is tossed and a dice is thrown.

Ans: A coin consists of two sides : head $\left( H \right)$ and tail $\left( T \right)$.

When a coin is tossed one time, the total number of outcomes are 2.

The probable outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

When a dice is thrown one time, the total number of outcomes are 6.

When a coin is tossed and dice is thrown then the total number of outcomes would be $2 \times 6$.

Thus the total number of outcomes when a coin is tossed and dice is thrown is 12.

Therefore the sample space are written as:

$S = \left\{ {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6} \right\}$

5. Describe the sample space for the indicated experiment: A coin is tossed and then a dice is thrown only in case a head is shown on the coin.

Ans: A coin consists of two sides : head $\left( H \right)$ and tail $\left( T \right)$.

When a coin is tossed one time, the total number of outcomes are 2.

The probable outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

When a dice is thrown one time, the total number of outcomes are 6.

When a coin is tossed and dice is thrown only in case a head is shown on the coin then the total number of outcomes would be $6 + 1$.

Thus the total number of outcomes when a coin is tossed and dice is thrown only in case a head is shown on the coin is 7.

Therefore the sample space are written as:

$S = \left\{ {H1,H2,H3,H4,H5,H6,T} \right\}$

6. 2 boys and 2 girls are in room X, and 1 boy and 3 girls are in room Y. Specify the sample space for experiment in which a room is selected and then a person.

Ans: Let 2 boys and 2 girls in room X be represented as ${B_1}$, ${B_2}$, ${G_1}$ and ${G_2}$ respectively.

Let 1 boy and 3 girls in room Y be represented as ${B_3}$, ${G_3}$, ${G_4}$ and ${G_5}$respectively.

Thus, the total number of outcomes in selecting a room and a person is 8.

Therefore the sample space are written as:

$S = \left\{ {X{B_1},X{B_2},X{G_1},X{G_2},Y{B_3},Y{G_3},Y{G_4},Y{G_5}} \right\}$

7. One die of red colour, one of white colour and one die of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space.

Ans: The probable outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

Let red, white, blue dies are represented by $R$, $W$ and $B$ respectively.

Thus the total number of outcomes in selecting a die and rolling it is 18.

Therefore the sample space are written as:

$S =\{ R1,R2,R3,R4,R5,R6,W1,W2,W3,W4,W5,W6, $

$ B1,B2,B3,B4,B5,B6 \}$

8. An experiment consists of recording boy-girl composition of families with 2 children.

(i). What is the sample space if we are interested in knowing whether it is a boy or a girl in the order of their births?

Ans: Let boy and girl be represented by $B$ and $G$ respectively.

Thus the total number of outcomes for knowing whether it's a boy or girl in their birth order is 4.

Therefore the sample spaces are written as:

$S = \left\{ {GG,GB,BG,BB} \right\}$

(ii). What is the sample space if we are interested in the number of girls in the family?

Ans: Let boy and girl be represented by $B$ and $G$ respectively.

It is given that there are two children in a family thus the total number of girls in one family can be 2 or 1 or 0.

Thus the total number of outcomes for knowing the number of girls in one family is 3.

Therefore the sample spaces are written as:

$S = \left\{ {0,1,2} \right\}$

9. A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

Ans: It is given that the box contains a total 4 balls, 1 red ball and 3 identical white balls.

Let the red ball and white ball are represented by $R$ and $W$ respectively.

Thus the total number of outcomes for drawing two successive balls without replacement is 3.

Therefore the sample spaces are written as:

$S = \left\{ {RW,WR,WW} \right\}$

10. An experiment consists of tossing a coin and then throwing it a second time if a head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the sample space.

Ans: A coin consists of two sides : head $\left( H \right)$ and tail $\left( T \right)$.

When a coin is tossed one time, the total number of outcomes are 2.

The probable outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

When a dice is thrown one time, the total number of outcomes are 6.

It is given that the coin is tossed again for the head that appears in tossing coin first and a dice is rolled if tail occurs in tossing coin first.

Thus the total number of outcomes would be 8.

Therefore the sample spaces are written as:

$S = \left\{ {HH,HT,T1,T2,T3,T4,T5,T6} \right\}$

11. Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective$\left( D \right)$ or non-defective$\left( N \right)$. Write the sample space of this experiment.

Ans: It is given that 3 bulbs are selected randomly from a lot of bulbs.

It is given that each bulb is classified as defective$\left( D \right)$ or non-defective$\left( N \right)$.

Therefore the sample spaces are written as:

$S = \left\{ {DDD,DDN,DND,DNN,NDD,NDN,NND,NNN} \right\}$

12. A coin is tossed. If the outcome is the head, a dice is thrown. If the die shows up an even number, the die is thrown again. What is the sample space for the experiment?

Ans: A coin consists of two sides : head $\left( H \right)$ and tail $\left( T \right)$.

When a coin is tossed one time, the total number of outcomes are 2.

The probable outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

When a dice is thrown one time, the total number of outcomes are 6.

It is given that a coin is tossed and if a head appears then a dice is thrown, if an even number appears on the dice the dice is rolled again.

Therefore sample spaces are written as:

$S =\{ T,H1,H3,H5,H21,H22,H23,H24,H25,H26, $

$ H41,H42,H43,H44,H45,H46,H61,H62, $

$ H63,H64,H65,H66\}$

13. The number 1, 2, 3 and 4 are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. A person draws two slips from the box, one after the another, without replacement. Describe the sample spaces for the experiment.

Ans: It is given that four slips are placed in a box numbered 1, 2, 3 and 4.

If 1 appears in the first slip, the three slips numbered 2, 3, 4 are left in the box. Similarly if 2 appears in first slip, the three slips numbered 1, 3, 4 are left in the box. Similarly for 3 and 4.

Therefore sample spaces are written as:

$S = \{ \left( {1,2} \right),\left( {1,3} \right),\left( {1,4} \right),\left( {2,1} \right),\left( {2,3} \right),\left( {2,4} \right) $

$ \left( {3,1} \right),\left( {3,2} \right),\left( {3,4} \right),\left( {4,1} \right),\left( {4,2} \right),\left( {4,3} \right) \}$

14. An experiment consists of rolling a die and then tossing a coin if the number on the die is even. If the number on the die is odd, the coin is tossed twice. Write the sample space for this experiment.

Ans: A coin consists of two sides : head $\left( H \right)$ and tail $\left( T \right)$.

When a coin is tossed one time, the total number of outcomes are 2.

The probable outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

When a dice is thrown one time, the total number of outcomes are 6.

It is given that a coin is tossed if the number appeared on the die is even and if the number appeared on the dice is odd then the coin is tossed twice.

Therefore sample space is written as:

$S =\{ 2H,2T,4H,4T,6H,6T,1HH,1HT,1TH,1TT $

$ 3HH,3HT,3TH,3TT,5HH,5HT,5TH,5TT \}$

15. A coin is tossed. If it shows a tail, we draw a ball from the box which contains 2 red and 3 black balls. If it showed head, we would throw a die. Find the sample space for this experiment.

Ans: A coin consists of two sides : head $\left( H \right)$ and tail $\left( T \right)$.

The probable outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

It is given that a box contains 2 red balls and 3 black balls.

Let denote 2 red balls by ${R_1}$, ${R_2}$ respectively and 3 black balls by ${B_1}$, ${B_2}$, ${B_3}$ respectively.

It is required to write sample space for the experiment, if the coin shows tail the ball is drawn from the bag and if head comes then dice is thrown.

Therefore sample space is written as:

$S = \left\{ {T{R_1},T{R_2},T{B_1},T{B_2},T{B_3},H1,H2,H3,H4,H5,H6} \right\}$

16. A die is thrown repeatedly until six comes up. What is the sample space for this experiment?

Ans: The probable outcomes when a dice is thrown are: 1, 2, 3, 4, 5 or 6.

It is required to write sample space for a die thrown repeatedly until six comes up.

It is possible the six comes up in first throw, but it is also possible the six does not comes in first throw the die is thrown again, it is also possible that six does not comes up even in second throw the dice is thrown third time and so on the process is repeated until six comes up.

Therefore sample space is written as:

$S =\{ 6,\left( {1,6} \right),\left( {2,6} \right),\left( {3,6} \right),\left( {4,6} \right),\left( {5,6} \right),\left( {1,1,6} \right), $

$ \left( {1,2,6} \right),....,\left( {1,5,6} \right),\left( {2,1,6} \right),\left( {2,2,6} \right),...., $

$ \left( {2,5,6} \right),....,\left( {5,1,6} \right),\left( {5,2,6} \right)...... \}$

## NCERT Solutions for Class 11 Maths Chapter 16 Probability Exercise 16.1

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