Question

Study the binomial distribution table. Notice that the probability of success on a single trial p ranges from 0.01 to 0.95. Some binomial distribution tables stop at 0.50 because of the symmetry in the table. Let's look for that symmetry. Consider the section of the table for which n = 5. Look at the numbers in the columns headed by p = 0.30 and p = 0.70. Do you detect any similarities? Consider the following probabilities for a binomial experiment with five trials.

(a) Compare P(3 successes), where p = 0.30, with P(2 successes), where p = 0.70.

P(3 successes), where p = 0.30, is smaller.They are the same.     P(3 successes), where p = 0.30, is larger.

(b) Compare P(3 or more successes), where p = 0.30, with P(2 or fewer successes), where p = 0.70.

P(3 or more successes), where p = 0.30, is smaller.They are the same.     P(3 or more successes), where p = 0.30, is larger.

(c) Find the value of P(4 successes), where p = 0.30. (Round your answer to three decimal places.)
P(4 successes) =  

For what value of r is P(r successes) the same using p = 0.70?
r =  

(d) What column is symmetrical with the one headed by p = 0.20?

the column headed by p = 0.85the column headed by p = 0.80     the column headed by p = 0.40the column headed by p = 0.50the column headed by p = 0.70

Answer 1

Answer:

a)

Given,

P(3 success) ; p = 0.3 ; n= 5

P(X = 3) = 5C3 * 0.3^3 * (1-0.3)^(5-3)

= 10*0.027*0.49

P(X = 3) = 0.1323

Now.

P(2 success) where p = 0.70

P(X = 2) = 5C2 * 0.7^2 * 0.3^3

= 10*0.49*0.027

P(X = 2) = 0.1323

Here we observe that both are same

b)

Now, P(3 or more) = P(X >= 3) ; p = 0.30

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

= 5C3 * 0.3^3 * (0.7)^2 + 5C4 * 0.3^4 * (0.7)^1 + 5C5 * 0.3^5 * (0.7)^0

= 0.1323 + 0.0283 + 0.0024

P(X >= 3) = 0.1630

Now P(2 or more) = P(X <= 2) ; p = 0.70

P(X <= 2) = P(X = 0) + P(X = 1) + P(X =2)

= 0.0024 + 0.0283 + 0.1323

P(X <= 2) = 0.1630

So here both are same

c)

P(4 successes), where p = 0.30

= 5C4*0.3^4 * (0.7)^1

P(4 successes) = 0.0285

For what value of r is P(r successes) the same using p = 0.70?

i.e.,

r = 1

It is due to P(X = 1) = 5C1*0.7^1*0.3^4

= 0.02835 for p = 0.70

d)

the column headed by p = 0.80  

Option B is right answer

Here by observing we can say that the column headed by p = 0.80 is symmetrical with one headed by p = 0.20