Question 1: Which of the following cannot be valid assignment of probability for elementary events or
outcomes of sample space

Elementary Events
(i)
(ii)
(iii)
(iv)

Answer:

For each event to be a valid assignment of probability, the probability of each event in sample space should be less than 1 and the sum of probability of all the events should be exactly equal to 1

(i) it is valid as each lies between 0 to 1 and sum of

(ii) it is valid as each lies between 0 to 1 and sum of

(iii) it is not valid as sum of which is greater than 1

(iv) it is not valid as which is greater than 1.

Question 2: A die is thrown. Find the probability of getting:
(i) a prime number           (ii) 2 or 4           (iii) a multiple of 2 or 3.

(iv) an even prime number          (v) a number greater than 5         

(vi) a number lying between 2 and 6

Answer:

(i) Prime numbers on a dice are 2, 3, and 5. So, the total number of prime numbers is 3. We know that, Probability = Number of favorable outcomes/ Total number of outcomes Thus, probability of getting a prime number

(ii) For getting 2 and 4, clearly the number of favorable outcomes is 2. We know that Probability = Number of favorable outcomes/ Total number of  

(iii) Multiple of 2 are 3 are 2, 3, 4 and 6. So, the number of favorable outcomes is 4 We know that, Probability = Number of favorable outcomes/ Total number 

(iv) An even prime number is 2 only. So, the number of favorable outcomes is 1. We know that, Probability = Number of favorable outcomes/ Total number of

(v) A number greater than 5 is 6 only. So, the number of favorable outcomes is 1. We know that, Probability = Number of favorable outcomes/ Total number of 

(vi) Total number on a dice is 6. Numbers lying between 2 and 6 are 3, 4 and 5 So, the total number of numbers lying between 2 and 6 is 3. We know that, Probability = Number of favorable outcomes/ Total number of outcomes Thus,

Question 3: In a simultaneous throw of a pair of dice, find the probability of getting:

(i) 8 as the sum     (ii) a doublet     (iii) a doublet of prime numbers     

(iv) a doublet of odd numbers     (v) a sum greater than 9     

(vi) an even number on first

(vii) an even number on one and a multiple of 3 on the other

(viii) neither 9 nor 11 as the sum of the numbers on the faces

(ix) a sum less than 6     (x) a sum less than 7     (xi) a sum more than 7

(xii) neither a doublet nor a total of 10     

(xiii) odd number on the first and 6 on the second

(xiv) a number greater than 4 on each die     (xv) a total of 9 or 11

(xvi) a total greater than 8

Answer:

In a throw of pair of dice, total no of possible outcomes which are

(i) Let E be event of getting the sum as 8

No. of favorable outcomes

(ii) Let E be the event of getting a doublet

No. of favorable outcomes

(iii) Let E be the event of getting a doublet of prime no’s

No. of favorable outcomes

(iv) Let E be the event of getting a doublet of odd no’s

No. of favorable outcomes

(v) E ⟶ event of getting a sum greater than 9

No. of favorable outcomes

(vi) Let E be the event of getting an even no. on first

No. of favorable outcomes

(vii) Let E be the event of getting an even no. on one and a multiple of 3 on other

No. of favorable outcomes

(viii) Let be the event of getting neither 9 nor 11 as the sum of numbers on faces

Therefore E is  getting either 9 or 11 as the sum of no’s on faces

No. of favorable outcomes

(ix) Let E be the event of getting a sum less than 6

No. of favorable outcomes

(x) Let E be the event of getting a sum less than 7

No. of favorable outcomes

(xi) Let E be the event of getting a sum more than 7

No. of favorable outcomes

(xii) Let E be the event of getting a 1 at least once

No. of favorable outcomes

(xiii) Let E be the event of getting a no other than 5 on any dice

No. of favorable outcomes

Question 4: In a single throw of three dice, find the probability of getting a total of 17 or 18.

Answer:

Three dices are thrown once.

Total number of possible outcomes is

Let E be the event of getting total of 17 or 18

Question 5: Three coins are tossed together. Find the probability of getting:
(i) exactly two heads        (ii) at least two heads       (iii) at least one head and one tail.

Answer:

Three coins are tossed together

Total number of possible outcomes is

Sample space for three coins tossed is 

(i) exactly two heads

(ii) at least two heads

(iii) at least one head and one tail.

Question 6: What is the probability that an ordinary year has 53 Sundays?

Answer:

Ordinary year has 365 days

365 days = 52 weeks + 1 day

That 1 day may be Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday

Total no. of possible outcomes

Let E be the event of getting 53 Sundays

No. of favorable outcomes = 1 {Sunday}

Question 7: What is the probability that a leap year has 53 Sundays and 53 Mondays?

Answer:

A leap year consists of 366 days comprising of 52 weeks and 2 days.

There are 7 possibilities for these 2 extra days viz.

(i) Sunday, Monday     (ii) Monday, Tuesday     (iii) Tuesday, Wednesday,

(iv) Wednesday, Thursday     (v) Thursday, Friday     (vi) Friday, Saturday and

(vii) Saturday, Sunday.

Let us consider two events :

A : the leap year contains 53 Sundays

B : the leap year contains 53 Mondays.

Hence, required probability

Question 8: A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that: (i) All the three balls are white.      (ii) A11 the three balls are red.        (iii) One ball is red and two balls are white

Answer:

A bag contains 8 red and 5 white balls.

(i) Probability when all the three balls are white

(ii) Probability when all the three balls are red 

(iii) Probability when One ball is red and two balls are white

Question 9: In a single throw of three dice, find the probability of getting the same number on all the three dice.

Answer:

Three dices are thrown once.

Total number of possible outcomes is

Let E be the event of getting same number on all the three dice

Question 10: Two unbiased dice are thrown. Find the probability that the total of the numbers on the dice is greater than 10.

Answer:

Two dices are thrown.

Total number of possible outcomes is

Let E be the event of getting sum on dice greater than 10

Question 11: A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn
15:

(i) a black king      (ii) either a black card or a king      (iii) black and a king

(iv) a jack, queen or a king      (v) neither a heart nor a king     

(vi) spade or an ace     (vii) neither an ace nor a king     

(viii) neither a red card nor a queen       (ix) other than an ace

(x) a ten      (xi) a spade      (xii) a black card      (xiii) the seven of clubs     

(xiv) jack      (xv) the ace of spades      (xvi) a queen      (xvii) a heart     

(xviii) a red card

Answer:

A card is drawn at random from a pack of 52 cards.

Total number of possible outcomes is

(i) a black king

Let E be the event of of getting a black king.

(ii) either a black card or a king

Let E be the event of of getting either a black card or a king

(iii) black and a king

Let E be the event of of getting either a black and a king

(iv) a jack, queen or a king

Let E be the event of of getting a jack, queen or a king

(v) neither a heart nor a king

Let E be the event of of getting neither a heart nor a king

(vi) spade or an ace     

Let E be the event of of getting spade or an ace

(vii) neither an ace nor a king

Let E be the event of of getting neither an ace nor a king

(viii) neither a red card nor a queen

Let E be the event of of getting neither a red card nor a queen

       

(ix) other than an ace

Let E be the event of of getting other than an ace

 

(x) a ten     

Let E be the event of of getting a ten

 

(xi) a spade     

Let E be the event of of getting a spade