Chapter 16: Probability Miscellaneous Exercise

In the question, it is given that,

Number of red marbles in the box = 10

Number of blue marbles in the box = 20

Number of green marbles in the box = 30

So, the total number of marbles in the box = 10 + 20 + 30 = 60

Number of ways of drawing 5 marbles from 60 marbles = ⁶⁰C₅

(i) All the drawn marbles will be blue if we draw 5 marbles out of 20 blue marbles.

We have,

Number of ways of drawing 5 blue marbles from 20 blue marbles = ²⁰C₅

Then,

The probability that all marbles will be blue = ²⁰C₅ / ⁶⁰C₅.

(ii) Number of ways in which the drawn marble is not green = ^(20+10)C₅

We have,

The probability that no marble is green = ³⁰C₅ / ⁶⁰C₅

Then,

The probability that at least one marble is green = 1 – ³⁰C₅ / ⁶⁰C₅.

In the question, it is given that,

4 cards are drawn from a well–shuffled deck of 52 cards

Number of ways of drawing 4 cards from 52 cards = ⁵²C₄

In a deck of 52 cards, there are 13 diamonds and 13 spades.

Number of ways of drawing 3 diamonds and one spade = ¹³C₃ × ¹³C₁

Therefore, the probability of obtaining 3 diamonds and one spade = (¹³C₃ × ¹³C₁)/ ⁵²C₄.

It is given that,

Die has two faces, each with the number ‘1’,

Three faces, each with the number ‘2’

And one face with the number ‘3’

Then, the total number of faces of the die = 6.

(i) P(2)

Number faces with number ‘2’ = 3 … [given]

So, P(2) = 3/6

= ½.

(ii) P(1 or 3)

We know that, P (1 or 3) = P (not 2) = 1 − P (2)

So, P(1 or 3) = 1 – ½

= (2 -1)/2

= ½.

(iii) P(not 3)

Number of faces with number ‘3’ = 1

P(3) = 1/6

P(not 3) = 1 – P(3)

= 1 – 1/6

= (6 – 1)/6

= 5/6.

It is given that,

Number of lottery tickets sold = 10,000

Number of prizes awarded = 10

(a) The probability of not getting a prize if we buy one ticket,

P (getting a prize) = 10/10000

= 1/1000

Then,

P (not getting a prize) = 1 – (1/1000)

= (1000 – 1)/1000

= 999/1000

Therefore, P (not getting a prize) = 999/1000.

(b) The probability of not getting a prize if we buy two tickets, then

Number of tickets not awarded = 10,000 − 10 = 9990

Therefore, P (not getting a prize) = ⁹⁹⁹⁰C₂ / ¹⁰⁰⁰⁰C_2.

(iii) The probability of not getting a prize if we buy 10 tickets, then

Number of tickets not awarded = 10,000 − 10 = 9990

Therefore, P (not getting a prize) = ⁹⁹⁹⁰C₁₀ / ¹⁰⁰⁰⁰C₁₀.

From the question,

Total number of students = 100

I and my friend are among the 100 students.

Then, two sections of 40 and 60 are formed.

Total number of ways of selecting 2 students out of 100 students =

(a) Let S = two of us will enter the same section if both of us are among 40 students or among 60 students.

Number of ways in which both of us enter the same section = P(S)

P(S) = (⁴⁰C₂ + ⁶⁰C₂) / ¹⁰⁰C₂

(b) P(we enter different sections)

= 1 − P(we enter the same section)

P(we enter different sections) = 1 – (17/33)

= (33 – 17)/33

= 16/33.

Let us assume L1, L2, and L3 be three letters and E1, E2, and E3 be their corresponding envelopes, respectively.

Then, the sample space is

L₁E₁, L₂E₃, L₃E₂,

L₂E₂, L₁E₃, L₃E₁,

L₃E₃, L₁E₂, L₂E₁,

L₁E₁, L₂E₂, L₃E₃,

L₁E₂, L₂E₃, L₃E₁,

L₁E₃, L₂E₁, L₃E₂,

Hence, there are 6 ways of inserting 3 letters in 3 envelopes.

And there are 4 ways in which at least one letter is inserted in the proper envelope. (first 4 rows of sample space)

The probability that at least one letter is inserted in the proper envelope,

= 4/6

= 2/3

It is given that A and B are two events, such that

P(A) = 0.54, P(B) = 0.69, P(A ∩ B) = 0.35

(i) P (A ∪ B)

We know that, P (A ∪ B) = P(A) + P(B) − P(A ∩ B)

⇒ P (A ∪ B) = 0.54 + 0.69 − 0.35 = 0.88

∴ P (A ∪ B) = 0.88.

(ii) P(A^I ∩ B^I)

We know that A^I ∩ B^I = (A ∪ B)^I [by De Morgan’s law]

So, P (A^I ∩ B^I) = P(A ∪ B)^I

= 1 − P(A ∪ B)

= 1 − 0.88 = 0.12

∴ P (A^I ∩ B^I) = 0.12.

(iii) P(A ∩ B^I)

We have,

P (A ∩ B^I) = P(A) − P(A ∩ B)

= 0.54 − 0.35

= 0.19

Therefore, P (A ∩ B^I) = 0.19.

(iv) P(B ∩ A^I)

We know that: P (B ∩ A^I) = P(B) − P(A ∩ B)

⇒ P (B ∩ A^I) = 0.69 – 0.35

Hence, P (B ∩ A^I) = 0.34.

From the given table,

The number of persons = 5

Out of 5 persons, 3 are Male

Out of 5 persons, 2 are 35 years of age.

Let us assume ‘A’ be the event in which the spokesperson will be a male and B be the event in which the spokesperson will be over 35 years of age.

Accordingly, P(A) = 3/5 and P(B) = 2/5

Since there is only one male who is over 35 years of age,

P(A ∩ B) = 1/5

We know that: P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

= (3/5) + (2/5) – (1/5)

= 4/5

Hence, the probability that the spokesperson will either be a male or over 35 years of age is 4/5.

(i) When the digits are repeated

Since four-digit numbers greater than 5000 are formed,

The thousand’s place digit is either 7 or 5.

Total number of 4-digit numbers greater than 5000 = 2 × 5 × 5 × 5 − 1

= 250 − 1

= 249

A number is divisible by 5 if the digit at its unit’s place is either 0 or 5.

∴ Total number of 4-digit numbers greater than 5000 that are divisible by 5

= 2 × 5 × 5 × 2 − 1

= 100 − 1 = 99

Hence, the probability of forming a number divisible by 5 when the digits are repeated is = P (number divisible by 5 when digits are repeated)

P(number divisible by 5 when digits repeated) = 99/249

= 33/83

(ii) When repetition of digits is not allowed

The thousands place can be filled with either of the two digits 5 or 7, i.e., in 2 ways.

The remaining 3 places can be filled with any of the remaining 4 digits.

Total number of 4-digit numbers greater than 5000 = 2 × 4 × 3 × 2 = 48

Here, the number of 4-digit numbers starting with 5 and divisible by 5

= 1 × 3 × 2 × 1 = 6

Here, the number of 4-digit numbers starting with 7 and divisible by 5

= 1 × 2 × 3 × 2 = 12

Total number of 4-digit numbers greater than 5000 that are divisible by 5

= 6 + 12 = 18

Thus, the probability of forming a number divisible by 5 when the repetition of digits is not allowed

= P (number divisible by 5 when digits are not repeated)

P (number divisible by 5 when digits are not repeated) = 18/48

= 3/8

It is given that,

The number lock has 4 wheels, each labelled with ten digits, i.e., from 0 to 9.

Then,

The number of ways of selecting 4 different digits out of the 10 digits = ¹⁰C₄

Now, each combination of 4 different digits can be arranged in 4! ways.

We have,

Number of four digits with no repetitions = 4! × ¹⁰C₄ = 5040

There is only one number that can open the suitcase.

Hence, the required probability is 1/5040.