Solution Found!
Suppose that a coin was definitely unbalanced and that the
Chapter 3, Problem 174E(choose chapter or problem)
Problem 174E
Suppose that a coin was definitely unbalanced and that the probability of a head was equal to p = .1. Follow instructions (a), (b), (c), and (d) as stated in Exercise 3.173. Notice that the probability distribution loses its symmetry and becomes skewed when p is not equal to 1/2.
Reference
A balanced coin is tossed three times. Let Y equal the number of heads observed.
a Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3.
b Construct a probability distribution similar to the one in Table 3.1.
c Find the expected value and standard deviation of Y, using the formulas E(Y ) = np and V (Y ) = npq.
d Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. Repeat for 2 standard deviations. How do your results compare with the results of Tchebysheff’s theorem and the empirical rule?
Questions & Answers
QUESTION:
Problem 174E
Suppose that a coin was definitely unbalanced and that the probability of a head was equal to p = .1. Follow instructions (a), (b), (c), and (d) as stated in Exercise 3.173. Notice that the probability distribution loses its symmetry and becomes skewed when p is not equal to 1/2.
Reference
A balanced coin is tossed three times. Let Y equal the number of heads observed.
a Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3.
b Construct a probability distribution similar to the one in Table 3.1.
c Find the expected value and standard deviation of Y, using the formulas E(Y ) = np and V (Y ) = npq.
d Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. Repeat for 2 standard deviations. How do your results compare with the results of Tchebysheff’s theorem and the empirical rule?
ANSWER:
Solution:
Step 1 of 4:
The coin was unbalanced and that the probability of a head was equal to p = 0.1
Let X follows the Binomial distribution with probability mass function
P(x) = , x = 0, 1, 2, ....
- The claim is to find the probabilities associated with X = 0, 1, 2, 3
Then, P( X = 0 ) =
= 0.729
P( X = 1) =
= 0.243
P( X = 2 ) =
= 0.027
P( X = 3 ) =
= 0.001
Hence, the P( X = 0 ) = 0.729 , P( X = 1 ) = 0.243, P( X = 2 ) = 0.027, and P( X = 3 ) = 0.001