Balbharati Maharashtra State Board 12th Commerce Maths Solution Book Pdf Chapter 8 Probability Distributions Ex 8.3 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 8 Probability Distributions Ex 8.3

Question 1.

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of (i) 2 successes (ii) at least 3 successes (iii) at most 2 successes.

Solution:

X: Getting an odd no.

p: Probability of getting an odd no.

A die is thrown 4 times

∴ n = 4

∵ p = \(\frac{3}{6}=\frac{1}{2}\)

∴ q = 1 – p = 1 – \(\frac{1}{2}\) = \(\frac{1}{2}\)

∵ X ~ B(3, \(\frac{1}{2}\))

∴ p(x) = \({ }^{n} \mathrm{C}_{x} p^{x} q^{n-x}\)

(i) P(Two Successes)

(ii) P(Atleast 3 Successes)

(iii) P(Atmost 2 Successes)

Question 2.

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes.

Solution:

n: No. of times die is thrown = 3

X: No. of doublets

p: Probability of getting doublets

Getting a doublet means, same no. is obtained on 2 throws of a die

There are 36 outcomes

No. of doublets are (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)

Question 3.

There are 10% defective items in a large bulk of items. What is the probability that a sample of 4 items will include not more than one defective item?

Solution:

n: No of sample items = 4

X: No of defective items

p: Probability of getting defective items

∴ p = 0.1

∴ q = 1 – p = 1 – 0.1 = 0.9

X ~ B(4, 0.1)

∴ p(x) = \({ }^{n} \mathrm{C}_{x} p^{x} \mathrm{q}^{n-x}\)

P(Not include more than 1 defective)

P(X ≤ 1) = p(0) + p(1)

= ^{4}C_{0} (0.1)^{0} (0.9)^{4} + ^{4}C_{1} (0.1)^{1} (0.9)^{4-1}

= 1 × 1 × (0.9)^{4} + 4 × 0.1 × (0.9)^{3}

= (0.9)^{3} [0.9 + 0.4]

= (0.9)^{3} × 1.3

= 0.977

Question 4.

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that (i) all the five cards are spades, (ii) only 3 cards are spades, (iii) none is a spade.

Solution:

X: No. of spade cards

Number of cards drawn

∴ n = 5

p: Probability of getting spade card

(i) P(All five cards are spades)

(ii) P(Only 3 cards are spades)

(iii) P(None is a spade)

Question 5.

The probability that a bulb produced by a factory will use fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.

Solution:

X : No. of bulbs fuse after 200 days of use

p : Probability of getting fuse bulbs

No. of bulbs in a sample

∴ n = 5

∴ p = 0.2

∴ q = 1 – p = 1 – 0.2 = 0.8

∵ X ~ B(5, 0.2)

∴ p(x) = \({ }^{n} \mathrm{C}_{x} p^{x} q^{n-x}\)

(i) P(X = 0) = ^{5}C_{0} (0.2)^{0} (0.8)^{5-0}

= 1 × 1 × (0.8)^{5}

= (0.8)^{5}

(ii) P(X ≤ 1) = p(0) + p(1)

= ^{5}C_{0} (0.2)^{0} (0.8)^{5-0} + ^{5}C_{1} (0.2)^{1} (0.8)^{5-1}

= 1 × 1 × (0.8)^{5} + 5 × 0.2 × (0.8)^{4}

= (0.8)^{4} [0.8 + 1]

= 1.8 × (0.8)^{4}

(iii) P(X > 1) = 1 – [p(0) + p(1)]

= 1 – 1.8 × (0.8)^{4}

(iv) P(X ≥ 1) = 1 – p(0)

= 1 – (0.8)^{5}

Question 6.

10 balls are marked with digits 0 to 9. If four balls are selected with replacement. What is the probability that none is marked 0?

Solution:

X : No. of balls drawn marked with the digit 0

n : No. of balls drawn

∴ n = 4

p : Probability of balls marked with 0.

∴ p = \(\frac{1}{10}\)

∴ q = 1 – p = 1 – \(\frac{1}{10}\) = \(\frac{9}{10}\)

p(x) = \({ }^{n} C_{x} p^{x} q^{n-x}\)

P(None of the ball is marked with digit 0)

Question 7.

In a multiple-choice test with three possible answers for each of the five questions, what is the probability of a candidate getting four or more correct answers by random choice?

Solution:

n: No. of Questions

∴ n = 5

X: No. of correct answers by guessing

p: Probability of getting correct answers

Question 8.

Find the probability of throwing at most 2 sixes in 6 throws of a single die.

Solution:

X : No. of sixes in 6 throws

n : No. of times dice thrown

∴ n = 6

p : Probability of getting six

∴ p = \(\frac{1}{6}\)

∴ q = 1 – p = 1 – \(\frac{1}{6}\) = \(\frac{5}{6}\)

∵ X ~ B(6, \(\frac{1}{6}\))

∴ p(x) = \({ }^{n} \mathrm{C}_{x} p^{x} q^{n-x}\)

P(At most 2 sixes)

P(X ≤ 2) = p(0) + p(1) + p(2)

Question 9.

Given that X ~ B(n, p),

(i) if n = 10 and p = 0.4, find E(X) and Var(X).

(ii) if p = 0.6 and E(X) = 6, find n and Var(X).

(iii) if n = 25, E(X) = 10, find p and Var(X).

(iv) if n = 10, E(X) = 8, find Var(X).

Solution:

∵ X ~ B (n, p), E(X) = np, V(X) = npq, q = 1 – p

(i) E(X) = np = 10 × 0.4 = 4

∵ q = 1 – p = 1 – 0.4 = 0.6

V(X) = npq = 10 × 0.4 × 0.6 = 2.4

(ii) ∵ p = 0.6

∴ q = 1 – p = 1 – 0.6 = 0.4

E(X) = np

∴ 6 = n × 0.6

∴ n = 10

∴ V(X) = npq = 10 × 0.6 × 0.4 = 2.4

(iii) E(X) = np

∴ 10 = 25 × p

∴ p = 0.4

∴ q = 1, p = 1 – 0.4 = 0.6

∴ S.D.(X) = √V(X)

= \(\sqrt{n p q}\)

= \(\sqrt{25 \times 0.4 \times 0.6}\)

= √6

= 2.4494

(iv) ∵ E(X) = np

∴ 8 = 10p

∴ p = 0.8

∴ q = 1 – p = 1 – 0.8 = 0.2

∵ V(X) = npq = 10 × 0.8 × 0.2 = 1.6