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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 11 Probability Distributions Ex 11.5

Question 1.

Compute P(X = k) for the binomial distribution, B(n, p) where

Solution:

Question 2.

The probability that Mr. Q hits a target at any trial is \(\frac{1}{4}\). Suppose he tries at the target 10 times. Find the probability that he hits the target

(i) exactly 4 times

(ii) at least one time.

Solution:

Let ‘p’ be the probability of hitting the trial

and number of trials ‘n’ = 10

Probability of ‘x’ success in ‘n’ trials is

(i) Probability that Mr.Q hits the target exactly 4 times is

(ii) Probability that Mr.Q hits the target atleast one time is

P(X ≥ 1) = 1 – P(X < 1)

= 1 – P(X = 0)

Question 3.

Using binomial distribution find the mean and variance of X for the following experiments

(i) A fair coin is tossed 100 times, and X denote the number of heads.

(ii) A fair die is tossed 240 times, and X denote the number of times that four appeared.

Solution:

(i) n = 100, ‘X’ denotes the number of heads.

(ii) n = 240, ‘X’ denotes the number of times four appeared.

Question 4.

The probability that a certain kind of component will survive a electrical test is \(\frac{3}{4}\). Find the probability that exactly 3 of the 5 components tested survive.

Solution:

Given n = 5

Question 5.

A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be

(i) at least one defective item

(ii) exactly two defective items.

Solution:

Given n = 10

Let ‘X’ be the random variable denotes the number of defective items.

∴ Probability of ‘x’ successes in ‘n’ trials is

Question 6.

If the probability that fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights.

(i) exactly 10 will have a useful life of at least 600 hours;

(ii) at least 11 will have a useful life of at least 600 hours;

(iii) at least 2 will not have a useful life of at least 600 hours.

Solution:

Given n = 12

Probability that a fluorescent light has a life of atleast of 600 hours is p = 0.9

∴ q = 1 – p = 1 – 0.9 = 0.1

Let ‘X’ be the number of bulbs.

∴ The probability of ‘x’ successes in ‘n’ trials is

(i) Probability that exactly 10 bulbs will have a useful life of atleast 600 hours

(ii) Probability that atleast 11 will have a useful life of atleast 600 hours is

(iii) Probability that atleast 2 will not have a useful life of 600 hours is

Question 7.

The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find

(i) the probability mass function

(ii) P(X = 3)

(iii) P(X ≥ 2) .

Solution:

Question 8.

If X ~ B(n, p) such that 4P(X = 4) = P(x = 2) and n = 6. Find the distribution, mean and standard deviation.

Solution:

Question 9.

In a binomial distribution consisting of 5 independent trials, the probability of 1 and 2 successes are 0.4096 and 0.2048 respectively. Find the mean and variance of the distribution.

solution:

Number of trials n = 5

The probability of ‘x’ successes in ‘n’ trials is

Given P(X = 1) = 0.4096 and P (X = 2) = 0.2048

Dividing Eq.(1) by Eq.(2)

### Samacheer Kalvi 12th Maths Solutions Chapter 11 Probability Distributions Ex 11.5 Additional Problems

Question 1.

In a Binomial distribution if n = 5 and P(X = 3) = 2P(X = 2) find p.

Solution:

Question 2.

If the sum of mean and variance of a Binomial Distribution is 4.8 for 5 trials find the distribution.

Solution:

np + npq = 4.8 ⇒ np (1 + q) = 4.8

5p [1 + (1 – p)] = 4.8

p^{2} – 2p + 0.96 = 0 ⇒ p = 1.2, 0.8

∴ p = 0.8 ; q = 0.2 [∵ p cannot be greater than 1]

∴ The Binomial distribution is P[X = x] = 5C_{x} (0.8)^{x} (0.2)^{5 – x}, x = 0 to 5

Question 3.

If on an average 1 ship out of 10 do not arrive safely to ports. Find the mean and the standard deviation of the ships returning safely out of a total of 500 ships.

Solution:

Probability of a ship arriving safely

Question 4.

The overall percentage of passes in a certain examination is 80. If six candidates appear in the examination what is the probability that at least five pass the examination.

Solution:

Pass percentage = 80%

∴ Probability of a candidate passing in the examination

Question 5.

In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is 5/6. What is the probability that he will knock down less than 2 hurdles?

Solution: