Mean and standard deviation of the binomial distribution

 

The mean and standard deviation for a binomial distribution are

 and ,

where

n-is the total number of trials,

 p-is the probability of success, and

q-is the probability of failure.

Example:

The probability that a certain rifleman will get a hit on any given shot at the rifle range is . If he fires one hundred shots, find the theoretical mean and standard deviation of x, the number of hits.

Solution:

We have  and .

Then, by formula, the mean is

and standard deviation is

              .

           

                               Exercises

1. Let x be a discrete random variable that possesses a binomial distribution. Using binomial formula, find the following probabilities:

a)  for n=8 and p=0.60

b)  for n=4 and p=0.30

c)  for n=6 and p=0.20

2. Determine the probability of getting:

       a) exactly three heads in 6 tosses of a fair coin;

       b) at least 3 heads in 6 tosses of a fair coin.

3. A card is drawn from an ordinary pack of playing cards, and its suit (clubs, diamonds, hearts, spades) noted, then it is replaced, the pack is shuffled, and another card is drawn. This is done until four cards have been drawn.

a) What is the probability that two spades will be drawn in four draws?

b) What is the probability that at least two spades will be drawn in four draws?

c) What is the probability that two red cards will be drawn in four draws?

d) What is the probability that at most two red cards will be drawn in four draws?

4. At a particular university it has been found that 20% of the students withdraw without completing the business statistics course. Assume that

20 students have registered for the course.

a) What is the probability that two or fewer will withdraw?

b) What is the probability that exactly four will withdraw?

c) What is the probability that more than three will withdraw?

d) What is the expected number and standard deviation of withdrawals?

5. For the binomial distribution with n=4 and p=0.25 find the probability of

       a) three or more successes

       b) at most three successes

       c) two or more failures.

6. Calculate the mean and standard deviation of the binomial distribution with         

       a) n=16;      p=0.5

       b) n=25       ;      p=0.1

       c) n=25;      p=0.9

7. 19% of cars in the country were at least 12 years old. Find the probability that in a random sample of 10 cars

a) exactly 4 are at least 12 years old;

b) exactly 2 are at least 12 years old;

c) none are at least 12 years old;

d) exactly 5 are at least 12 years old.

8. Suppose that, for a particular type of a cancer, treatment provides

a 5-or more years survival rate of 80% if the disease could be detected at an early stage. Among 18 patients diagnosed to have this form of cancer at an early stage who are just starting this treatment, find the probability that

a) fourteen will survival beyond 5-years;

b) six will die within 5-years;

c) the number of patients surviving beyond 5-years will be between 9 and 13(inclusive);

d) find the expectation and standard deviation of the number of 5-years survivors.

9. It is known that 3% of produced goods have some defects. Eight of these goods are selected randomly.

a) What is the probability that none of these goods are defective?

b) What is the probability that one of these goods is defective?

c) What is the probability that at least two of these goods are defective?

10. A certain type of infection is spread by contact with an infected person. Let the probability that a healthy person gets the infection, in one contact,

be p=0.4.

a) An infected person has contact with five healthy persons. Specify the distribution of X = number of persons who contact the infection.

b) Find ; ; and E[X].

11. A salesman of home computers will contact four customers during a week. Each contact can result in either a sale, with probability 0.20, or no sale with probability 0.80. Assume that customer contacts are independent. Let X denotes the number of computers sold during the week.

a) Obtain the probability distribution of X.

b) Calculate the expected value of X.

 

                          Answers

1. a) 0.279; b) 0.076; c) 0.246; 2. a) 5/16; b) 21/32; 3. a) 0.211; b) 0.262;

c) 0.375; d) 0.688; 4. a) 0.2060; b) 0.2182; c) 0.5886; d) 4; 1.790;

5. a) 0.051; b) 0.996; c) 0.949; 6. a) 8;2; b) 2.5; 1.5; c) 22.5; 1.5;

7. a) 0.0773; b) 0.3010; c) 0.1216; d) 0.0218; 8. a) 0.215; b) 0.082; c) 0.283;

d) 14.4; 1.697; 9. a) 0.784; b) 0.194; c) 0.022; 10. a) binomial distribution with n=5; p=0.4; b) 0.913; 0.078; 2; 11. a) P(0)=0.4096; P(1)=0.4096; P(2)=0.1536; P(3)=0.0256; P(4)=0.0016; b) 0.8;

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