Class 12 RD Sharma Solutions – Chapter 33 Binomial Distribution – Exercise 33.1 | Set 1

Question 1. There are 6% defective items in a large bulk of items. Find the probability that a sample of 8 items will include not more than one defective item.

Solution:

Let us consider X be the number of defective items in a sample of 8 items. 

So, a binomial distribution follows by x with n = 8.

and Probability of getting a defective item(p) = 0.06 

The probability of getting a defective item (1 – p) = 0.94

So, P(X = r) = ⁸C_r (0.06)^r (0 . 94 )^{8 – r}, where r = 0, 1, 2, 3, . . . 8

So, the required probability is

= P (X < 1) 

= P(X = 0) + P(X = 1)

= ⁸C₀ (0.06)⁰ (0.94)^8-0 + ⁸C₁ (0.06)¹ (0.94 )^8-1

= (0.94)⁸ + 8 (0.06) (0.94)⁷ 

= (0.94)⁷ {0.94 + 0.48}

= 1.42 (0.94)⁷

Question 2. A coin is tossed 5 times. What is the probability of getting at least 3 heads?

Solution:

Let us consider X be the number of heads in 5 tosses.

So, a binomial distribution follows by x with n = 5

Probability of getting a head(p) = 1/2

And q = 1 – p = 1 – 1/2 = 1/2

P(X = r) = ⁵C_r (1/2)^r (1/2)^5-r , where r = 0, 1, 2 . . . 5

So, the required probability is

= P(X > 3)

= P(X = 3) + P(X = 4) + P(X = 5)

= ⁵C₃ (1/2)³ (1/2)^5-3 + ⁵C₄ (1/2)¹ (1/2)^5-1 + ⁵C₅ (1/2)⁵ (1/2)^5-0

= 10 (1/2)⁵ + 5 (1/2)⁵ + 1 (1/2)⁵

= (1/2)⁵ (10 + 5 + 1)

= (1/2)⁵ (16)

= 1/2

Question 3. A coin is tossed 5 times. What is the probability that tail appears an odd number of times?

Solution:

Let us consider X be the number of tails when a coin is tossed 5 times.

So, a binomial distribution follows by x with n  = 5

Probability of getting head(p) = 1/2.

Also, q = 1 – p = 1/2

P(X = r) = ⁵C₃ (1/2)^r (1/2)^n-r = ⁵C_r (1/2)⁵

So, the required probability is

= P(X = 1) + P(X = 3) + P(X = 5)

= ⁵C₁ (1/2)⁵ + ⁵C₃ (1/2)⁵ + ⁵C₅ (1/2)⁵

= (1/2)⁵ [5 + 10 + 1]

= 16/32

= 1/2

Question 4. A pair of dice is thrown 6 times. If getting a total of 9 is considered a success, what is the probability of at least 5 successes?

Solution:

Let us consider X be the number of successes in 6 throws of the two dice.

So the probability of success is equal to the probability of getting a total of 9  

= Probability of getting (3, 6), (4, 5), (5, 4), (6, 3) from 36 outcomes

Here, p = 4/36 = 1/9

Also q = 1 – p = 8/9 and n = 6

Now, a binomial distribution X follows with n = 6, p = 1/9, and q = 8/9

P(X = r) = ⁶C_r (1/9)^r (8/9)^6-r

The required probability is

= P(X > 5)

= P(X = 5) + P(X = 6)

= 

= 

= 49/9⁶

Question 5. A fair coin is tossed 8 times, find the probability: 

(i) of exactly 5 heads.

Solution:

Let us consider X be the number of heads obtained when a fair is tossed 8 times.

Now, a binomial distribution X follows with n  = 8.

Here, p = 1/2 and q = 1 – 1/2 = 1/2

So, P(X = r) = 

= ⁸C_r (1/2)⁸ , where r = 0, 1, 2, . . . , 8

So, the probability of obtaining exactly 5 heads is 

P(X = 5) = ⁸C₅ (1/2)⁸

= 56/256

= 7/32

(ii) of at least six heads.

Solution:

Let us consider X be the number of heads obtained when a fair is tossed 8 times.

Now, a binomial distribution X follows with n = 8.

Here, p = 1/2 and q = 1 – 1/2 = 1/2

So, P(X = r) = 

= ⁸C_r (1/2)⁸ , where r = 0, 1, 2, . . . , 8

So, the probability of obtaining exactly 5 heads is 

P(X = 5) = ⁸C₅ (1/2)⁸

= 56/256

= 7/32

(iii) of at most six heads.

Solution:

Let us consider X denotes the number of heads obtained when a fair is tossed 8 times.

Now, a binomial distribution X follows with n = 8.

Here p = 1/2 and q = 1 – 1/2 = 1/2

P(X = r) = 

= ⁸C_r (1/2)⁸ , r = 0, 1, 2, . . . , 8

so the probability of obtaining at most 6 heads is

P (X < 6) = 1 – [P(X = 7) + P(X = 8)]

= 

= 1 – (8/256 + 1/256)

= 1 – 9/256

= 247/256

Question 6. Find the probability of 4 turning up at least once in two tosses of a fair die.

Solution:

Let us consider X denotes the probability of getting 4 in two tosses of a fair die.

Now, a binomial distribution X follows with n = 2.

Here, p = 1/6 and q = 5/6

P(X = r) = 

So, the probability of obtaining 4 at least once is

P(X > 1) = 1 – P(X = 0)

= 1 – 

= 1 – 25/36

= 11/36

Question 7. A coin is tossed 5 times. What is the probability that head appears an even number of times?

Solution:

Let us consider X denotes the number of heads that appear when a coin is tossed 5 times.

Now, a binomial distribution X follows with n = 5 

Here p = q = 1/2.

P(X = r) = 

= ⁵C_r (1/2)⁵

P (head appears an even number of  times) = P(X = 0) + P(X = 2) + P(X = 4)

= 

= 

= 16/32

= 1/2

Question 8. The probability of a man hitting a target is 1/4. If he fires 7 times, what is the probability of his hitting the target at least twice?

Solution:

Let us consider X denotes the number of times the target is hit. 

Now, a binomial distribution X follows with n = 7.

Here, p = 1/4 and q = 3/4.

P(X = r) = 

P(hit the target at least twice) = P(X > 2) 

= 1 – {P(X = 0) + P(X = 1)}

= 

= 1 – (3/4)⁷ – 7 (1/4) (3/4)⁶

= 1 – 1/16384(2187 + 5103)

= 1 – 3645/8192

= 4547/8192

Question 9. Assume that on average one telephone number out of 15 called between 2 P.M. and 3 P.M. on weekdays is busy. What is the probability that if six randomly selected telephone numbers are called, at least 3 of them will be busy?

Solution:

Let us consider X denotes the number of busy calls for 6 randomly selected telephone numbers.

Now, a binomial distribution X follows with n = 6.

Here, p = one out of 15 = 1/15

And q = 1 – 1/15 = 14/15

P(X = r) = 

So, the probability that at least 3 of them are busy is 

P(X > 3) = 1 – {P(X = 0) + P(X = 1) + P(X = 2)}

= 

= 

Question 10. If getting 5 or 6 in a throw of an unbiased die is a success and the random variable X denotes the number of successes in six throws of the die, find P (X ≥ 4).

Solution:

Let us consider X be the number of successes. That is getting 5 or 6 in a throw of die in 6 throws.

Now, a binomial distribution X follows with n = 6

Here, p = of getting 5 or 6 = 1/6 + 1/6 = 1/3

 And q = 1 – p = 2/3

P(X = r) = 

P(X > 4) = P(X = 4) + P(X = 5) + P(X = 6)

= 

= (1/3⁶) (60 + 12 + 1)

= 73/729

Question 11. Eight coins are thrown simultaneously. Find the chance of obtaining at least six heads.

Solution:

Let us consider X denotes the number of heads in tossing 8 coins.

Now, a binomial distribution X follows with n = 8.

Here, p = 1/2 and q = 1/2

P(X = r) = 

= ⁸C_r (1/2)⁸

So, the probability of getting at least 6 heads is 

P(X > 6) = P(X = 6) + P(X = 7) + P(X = 8)

= 

= 1/2⁸(28 + 8 + 1)

= 37/256

Question 12. Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards.

(i) What is the probability that all the five cards are spades?

Solution:

Let us consider X be the number of spade cards when 5 cards are drawn with replacement.

Now, a binomial distribution X follows with n = 5.

p = 13/52 = 1/4 and q = 1 – p = 3/4

P(X = r) = 

P(All cards are spades) = P(X = 5)

= 

= 1/1024

(ii) What is the probability that only 3 cards are spades? 

Solution:

Let us consider X be the number of spade cards when 5 cards are drawn with replacement.

Now, a binomial distribution X follows with n = 5. 

Here, p = 13/52 = 1/4 and q = 1 – p = 3/4

P(X = r) = 

P(only 3 cards are spades) = P(X = 3)

= 

= (1/1024) (90)

= 45/512

(iii) What is the probability that none is a spade?

Solution:

Let us consider X be the number of spade cards when 5 cards are drawn with replacement.

Now, a binomial distribution X follows with n = 5.

Here p = 13/52 = 1/4 and q = 1 – p = 3/4.

P(X = r) = 

P(none is a spade) = P(X = 0)

= 

= 243/1024

Question 13. A bag contains 7 red, 5 white, and 8 black balls. Four balls are drawn one by one with replacement.

(i) What is the probability that none is white?

Solution:

Let us consider X denotes the number of white balls drawn when 4 balls are drawn with replacement.  

Now, a binomial distribution X follows with n = 4.

Here, the probability for a white ball(p) = 5/20 = 1/4

And q = 1 – p = 3/4

P(X = r) = 

So, the probability that none is white is

P(X = 0) = 

= 81/256

(ii) What is the probability that none is white?

Solution:

Let us consider X denotes the number of white balls drawn when 4 balls are drawn with replacement.  

Now, a binomial distribution X follows with n = 4.

Here, the probability for a white ball(p) = 5/20 = 1/4

And q = 1 – p = 3/4

P(X = r) = 

so, the probability that none is white is 

P(X = 0) = 

= 81/256

(iii) What is the probability that any two are white?

Solution:

Let us consider X denotes the number of white balls drawn when 4 balls are drawn with replacement.  

Now, a binomial distribution X follows with n = 4.

So, the probability for a white ball(p) = 5/20 = 1/4

And q = 1 – p = 3/4

P(X = r) = 

So, the probability that any two are white is 

P(X = 2) = 

= 54/256

= 27/128

Question 14. A box contains 100 tickets, each bearing one of the numbers from 1 to 100. If 5 tickets are drawn successively with replacement from the box, find the probability that all the tickets bear numbers divisible by 10.

Solution:

Let us consider X denotes the variable representing number on the ticket bearing a number divisible by 10 out of the 5 tickets drawn.

Now, a binomial distribution X follows with n = 5.

Here, the probability of getting a ticket bearing number divisible by 10(p) = 10/100

= 1/10

And q = 1 – p = 9/10

P(X = r) = 

So, the probability that all the tickets bear numbers divisible by 10 is

P(X = 5) = 

= (1/10)⁵ (9/10)⁰

= (1/10)⁵

Question 15. A bag contains 10 balls, each marked with one of the digits from 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?

Solution:

Let us consider X denotes the number of balls marked with the digit 0 when 4 balls are drawn successfully with replacement.

Now, a binomial distribution X follows with n = 4.

So, the probability that a ball randomly drawn bears digit 0(p) = 1/10

And q = 1 – p = 9/10

P(X = r) = 

P(none bears the digit 0) = P(X = 0)

= 

= (9/10)⁴

Question 16. In a large bulk of items, 5 percent of the items are defective. What is the probability that a sample of 10 items will include not more than one defective item?

Solution:

Let us consider X be the number of defective items in a sample of 10 items.

Now, a binomial distribution X follows with n = 10.

And the probability of defective items(p) = 5% = 0.05 

And q = 1 – p = 0.95

P(X = r) = ¹⁰C_r (0. 05)^r (0.95)^10-r

So, the probability(sample of 10 items will include not more than one defective item) is 

P(X < 1) = P(X = 0) + P(X = 1) 

= 10C0 (0.05)⁰ (0.95 )^{0-0 + ¹⁰C₁ (0.05)¹ (0.95)^10-1

= (0.95)⁹ (0.95 + 0.5)

= 1.45 (0.95)⁹

Question 17. The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. 

(i) Find the probability that out of 5 such bulbs none will fuse after 150 days of use. 

Solution:

Let us consider X be the number of bulbs that fuse after 150 days.

Now, a binomial distribution X follows with n = 5.

Here, p = 0.05 and  q = 0.95

Or p = 1/20 and q = 19/20

P(X = r) = 

So, the probability (none will fuse after 150 days of use) is 

P(X = 0) = 

= (19/20)⁵

(ii) Find the probability that out of 5 such bulbs, not more than one will fuse after 150 days of use. 

Solution:

Let us consider X denotes the number of bulbs that fuse after 150 days.

Now, a binomial distribution X follows with n = 5.

Here p = 0.05 and q = 0.95.

Or p = 1/20 and q = 19/20

P(X = r) = 

So, the probability (not more than 1 will fuse after 150 days of use) is

P(X < 1) = P(X = 0) + P(X = 1)

= 

= 

= (6/5) (19/20)⁴

(iii) Find the probability that out of 5 such bulbs more than one will fuse after 150 days of use 

Solution:

Let us consider X denotes the number of bulbs that fuse after 150 days.

Now, a binomial distribution X follows with n = 5.

Here p = 0.05 and q = 0.95

Or p = 1/20 and q = 19/20

P(X = r) = 

So, the probability(more than one will fuse after 150 days of use) is

P(X > 1) = 1 – P(X < 1)

= 1 – (6/5) (19/20)⁴

(iv) Find the probability that out of 5 such bulbs at least one will fuse after 150 days of use. 

Solution:

Let us consider X denotes the number of bulbs that fuse after 150 days.

Now, a binomial distribution X follows with n = 5.

Here p = 0.05 and q = 0.95

Or p = 1/20 and q = 19/20

P(X = r) = 

So, the probability(at least one will fuse after 150 days of use) is

P(X > 1) = 1 – P(X = 0)

= 1 – (19/20)⁵

Question 18. Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?

Solution:

Let us consider X denotes the number of people that are right-handed in the sample of 10 people.

Now, a binomial distribution X follows with n = 10.

Here p = 90 % = 90/100 = 0.9

And q = 1 – p = 0.1

P(X = r) = ¹⁰C_r (0.9)^r (0.1)^10-r

So, the probability that at most 6 are right – handed is

P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)

= 1 – {P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)}

=