(a) \(P( X = 5) \approx 0.273\)

(b) \(P( X \geq 6) \approx 0.753\)

(c) \(P( X < 4) \approx 0.016\)

UpStudy Free Solution:

To solve these problems, we will use the binomial probability formula. The binomial probability formula is given by:

\[P( X = k) = \binom { n} { k} p^ k ( 1- p) ^ { n- k} \]

where:

- \(n\) is the number of trials (in this case, 10),

- \(k\) is the number of successes (the number of U.S. adults with very little confidence in newspapers),

- \(p\) is the probability of success on a single trial (in this case, 0.63),

- \(\binom { n} { k} \) is the binomial coefficient, calculated as \(\binom { n} { k} = \frac { n! } { k! ( n- k) ! } \).

Part (a): Exactly Five

We need to find \(P( X = 5) \):

\[P( X = 5) = \binom { 10} { 5} ( 0.63) ^ 5 ( 1- 0.63) ^ { 10- 5} \]

Calculate the binomial coefficient:

\[\binom { 10} { 5} = \frac { 10! } { 5! 5! } = 252\]

Now calculate the probability:

\[P( X = 5) = 252 \times ( 0.63) ^ 5 \times ( 0.37) ^ 5\]

Using a calculator:

\[( 0.63) ^ 5 \approx 0.156\]

\[( 0.37) ^ 5 \approx 0.007\]

So,

\[P( X = 5) \approx 252 \times 0.156\times 0.007 \approx 252 \times 0.00108 = 0.27260\]

Rounded to three decimal places:

\[P( X = 5) \approx 0.273\]

Part (b): At Least Six

We need to find \(P( X \geq 6) \). This is the sum of the probabilities for \(X = 6, 7, 8, 9,\) and \(10\):

\[P( X \geq 6) = P( X = 6) + P( X = 7) + P( X = 8) + P( X = 9) + P( X = 10) \]

We will use the binomial formula for each term:

\[P( X = 6) = \binom { 10} { 6} ( 0.63) ^ 6 ( 0.37) ^ 4\]

\[P( X = 7) = \binom { 10} { 7} ( 0.63) ^ 7 ( 0.37) ^ 3\]

\[P( X = 8) = \binom { 10} { 8} ( 0.63) ^ 8 ( 0.37) ^ 2\]

\[P( X = 9) = \binom { 10} { 9} ( 0.63) ^ 9 ( 0.37) ^ 1\]

\[P( X = 10) = \binom { 10} { 10} ( 0.63) ^ { 10} ( 0.37) ^ 0\]

Using a calculator or binomial probability table/software, we can find these values:

\[P( X = 6) \approx 0.231\]

\[P( X = 7) \approx 0.275\]

\[P( X = 8) \approx 0.180\]

\[P( X = 9) \approx 0.061\]

\[P( X = 10) \approx 0.006\]

Summing these probabilities:

\[P( X \geq 6) \approx 0.231 + 0.275 + 0.180 + 0.061 + 0.006 = 0.753\]

Rounded to three decimal places:

\[P( X \geq 6) \approx 0.753\]

Part (c): Less than Four

We need to find \(P( X < 4) \). This is the sum of the probabilities for \(X = 0, 1, 2,\) and \(3\):

\[P( X < 4) = P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) \]

We will use the binomial formula for each term:

\[P( X = 0) = \binom { 10} { 0} ( 0.63) ^ 0 ( 0.37) ^ { 10} \]

\[P( X = 1) = \binom { 10} { 1} ( 0.63) ^ 1 ( 0.37) ^ 9\]

\[P( X = 2) = \binom { 10} { 2} ( 0.63) ^ 2 ( 0.37) ^ 8\]

\[P( X = 3) = \binom { 10} { 3} ( 0.63) ^ 3 ( 0.37) ^ 7\]

Using a calculator or binomial probability table/software, we can find these values:

\[P( X = 0) \approx 0.000\]

\[P( X = 1) \approx 0.000\]

\[P( X = 2) \approx 0.002\]

\[P( X = 3) \approx 0.014\]

Summing these probabilities:

\[P( X < 4) \approx 0.000 + 0.000 + 0.002 + 0.014 = 0.016\]

Rounded to three decimal places:

\[P( X < 4) \approx 0.016\]

Supplemental Knowledge

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. It is widely used in various fields such as statistics, finance, and science to model scenarios where there are two possible outcomes (success or failure).

Key Concepts:

1. Binomial Coefficient (\(\binom { n} { k} \)):

The binomial coefficient represents the number of ways to choose \(k\) successes out of \(n\) trials and is calculated as:

\[\binom { n} { k} = \frac { n! } { k! ( n- k) ! } \]

where \(n! \) (n factorial) is the product of all positive integers up to \(n\).

2. Probability Mass Function (PMF):

The PMF of a binomial distribution gives the probability that there are exactly \(k\) successes in \(n\) trials:

\[P( X = k) = \binom { n} { k} p^ k ( 1- p) ^ { n- k} \]

Here, \(p\) is the probability of success on a single trial, and \(1- p\) is the probability of failure.

3. Cumulative Probability:

To find the probability of having at least or less than a certain number of successes, we sum up individual probabilities:

- At least \(k\):

\[P( X \geq k) = P( X = k) + P( X = k+ 1) + ... + P( X = n) \]

- Less than \(k\):

\[P( X < k) = P( X = 0) + P( X = 1) + ... + P( X = k- 1) \]

4. Applications:

- Quality control in manufacturing.

- Clinical trials in medicine.

- Survey analysis in social sciences.

Understanding these concepts helps you tackle problems involving binomial distributions effectively.