5 girls and 10 boys sit at random in a row having 15 chairs numbered as 1 to 15 , then find the probability that end seats are occupied by the girls and between any two girls an odd number of boys sit-

Total number of arrangements = 15 ! Extreme chairs are occupied by girls, thus there are four gaps among 5 girls where boys can be seated. Let the number of boys in these four gaps be 2 x + 1 , 2 y + 1 , 2 z + 1 and 2 t + 1 , then 2 x + 1 + 2 y + 1 + 2 z + 1 + 2 t + 1 = 10 ⇒ x + y + z + t = 3 where x , y , z , t are integers and 0 ≤ x ≤ 3 , 0 ≤ y ≤ 3 , 0 ≤ z ≤ 3 , 0 ≤ t ≤ 3 Therefore, total number of ways of selecting positions for boys = coefficient of x 3 in 1 + x + x 2 + x 3 4 = coefficient of x 3 in 1 - x 4 1 - x 4 = coefficient of x 3 in 1 - x 4 4 ( 1 - x ) - 4 = coefficient of x 3 in 1 - 4 x 4 + 6 x 8 - 4 x 12 + . . . 1 + 4 x + 10 x 2 + 20 x 3 + . . . = 20 ∴ Number of arrangements of boys and girls with given condition = 20 × 10 ! × 5 ! ∴ Required probability = 20 × 10 ! × 5 ! 15 ! .

A dice has one 1 , two 2 ' s and three 3 ' s on its faces. A player throws it till he gets three consecutive 1 ' s . If p n is the probability that no 3 consecutive 1 ' s appear in n throws, then prove that (i) p 1 = p 2 = 1 and p 3 = 215 216

We have a dice with outcomes 1 , 2 , 2 , 3 , 3 , 3 P 1 = 1 6 , P 2 = 2 6 = 1 3 , P 3 = 3 6 = 1 2 Now, if we throw a dice once, then there is way of getting three consecutive 1 ' s , hence p 1 = 1 Similarly, p 2 = 1 . Now, probability of getting 1 in three consecutive throws = 1 6 × 1 6 × 1 6 = 1 216 Hence, p 3 = 1 - 1 216 = 215 216 .

Let A and B be two events such that P A ∪ B = 1 6 , P A ∩ B = 1 4 , and P A = 1 4 , where A ¯ stands for complement of event A . Then events A and B are

independent but not equally likely.

mutually exclusive and independent.

equally likely but not independent.

equally likely and mutually exclusive.

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A Sudoku matrix is defined as a $9 \times 9$ array with entries from ${1, 2, 3 \dots . . . 9}$ and with the constraint that each row, each column and each of the nine $3 \times 3$ boxes that tile the array contains each digit from $1$ to $9$ exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has an equal probability of being chosen). We know two squares in this matrix as shown. Then find the probability that the square marked by ? contains the digit $3 .$

Important Questions on Probability

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Alpha Question Bank for Engineering: Mathematics>Chapter 10 - Probability>Exercise-4>Q 4

5

girls and

10

boys sit at random in a row having

15

chairs numbered as

1

15,

then find the probability that end seats are occupied by the girls and between any two girls an odd number of boys sit-

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Alpha Question Bank for Engineering: Mathematics>Chapter 10 - Probability>Exercise-4>Q 5

Team

A

plays with

5

other teams exactly once. Assuming that for each match the probabilities of a win, draw and loss are equal then find the probability that

A

wins and losses equal no. of matches.

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Alpha Question Bank for Engineering: Mathematics>Chapter 10 - Probability>Exercise-4>Q 6

Suppose that

S

be the set of all the ordered

4

-tuples

(x, y, z, w)

of the

+ ve

integers, which are the solutions of

x + y + z + w = 21

. One such ordered tuple of solution is selected at random from

S .

Then find the probability that

x > y .

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Alpha Question Bank for Engineering: Mathematics>Chapter 10 - Probability>Exercise-4>Q 7

In a betting game in an exhibition two dice

P

and

Q

are being used. Dice

P

has four red faces and two white faces, whereas dice

Q

has two red and four white faces. A fair coin is tossed once. If it shows head the game continues by throwing dice

P .

If it falls tail dice

Q

is thrown. If first

n

throws of the die all turns up red, then find the probability that

P

is being used.

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Alpha Question Bank for Engineering: Mathematics>Chapter 10 - Probability>Exercise-4>Q 8

On a particular day, six persons pick six different books, one each from different counters at a public library. At the closing time, they arbitrarily put their books to the vacant counters. Then find the probability that exactly two books are at their previous places.

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Alpha Question Bank for Engineering: Mathematics>Chapter 10 - Probability>Exercise-4>Q 9

A dice has one

1,

two

2' s

and three

3' s

on its faces. A player throws it till he gets three consecutive

1' s .

p_{n}

is the probability that no

3

consecutive

1' s

appear in

n

throws, then prove that
(i)

p_{1} = p_{2} = 1

and

p_{3} = \frac{215}{216}

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Alpha Question Bank for Engineering: Mathematics>Chapter 10 - Probability>Exercise-4>Q 11

n

students filled their forms for a competitive exam. Probability that exactly

r

students will not appear in the exam is proportional to

r

. If probability that out of remaining

n - r

students exactly

i

students are selected is proportional to

i

. Prove that the probability of two students finally selected is

\frac{8}{n (n + 1)} [n (\frac{1}{2} - \frac{1}{n}) - (\frac{1}{3} + \frac{1}{4} + \dots \dots + \frac{1}{n})]

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Alpha Question Bank for Engineering: Mathematics>Chapter 10 - Probability>Exercise-4>Q 12

In an organization number of women are

μ

times that of men. If

n

things are to be distributed among them then the probability that the number of things received by men are odd is

(\frac{1}{2} - {(\frac{1}{2})}^{n + 1})

. Evaluate

μ .

Important Questions on Probability

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