Questions: 5 points Binomial Distributions: The National Institute of Mental Health reports that there is a 20% chance of an adult American suffering from a psychiatric disorders. Seven randomly selected adult Americans are examined for psychiatric disorders. A success is defined as an adult American having a psychiatric disorder. The random variable X, is the number of adult Americans, out of 7 adult Americans, who have a psychiatric disorder. Using the binomial table provided to you, what is the probability that exactly 1 out of 7 adult Americans will have a psychiatric disorder? Do not round. 0.3670

5 points
Binomial Distributions:
The National Institute of Mental Health reports that there is a 20% chance of an adult American suffering from a psychiatric disorders.
Seven randomly selected adult Americans are examined for psychiatric disorders.
A success is defined as an adult American having a psychiatric disorder.
The random variable X, is the number of adult Americans, out of 7 adult Americans, who have a psychiatric disorder.

Using the binomial table provided to you, what is the probability that exactly 1 out of 7 adult Americans will have a psychiatric disorder? Do not round.
0.3670

Transcript text: 14 5 points $\square$ Binomial Distributions: The National Institute of Mental Health reports that there is a 20\% chance of an adult American suffering from a psychiatric disorders. Seven randomly selected adult Americans are examined for psychiatric disorders. A success is defined as an adult American having a psychiatric disorder. The random variable $X$, is the number of adult Americans, out of 7 adult Americans, who have a psychiatric disorder. Using the binomial table provided to you, what is the probability that exactly 1 out of 7 adult Americans will have a psychiatric disorder? Do not round. \[ 0.3670 \] Previous Next

Solution

Solution Steps

To solve this problem, we need to use the binomial probability formula. The binomial distribution is appropriate here because we have a fixed number of trials (7 adult Americans), two possible outcomes (having or not having a psychiatric disorder), and a constant probability of success (20% or 0.2). We want to find the probability of exactly 1 success (psychiatric disorder) out of 7 trials.

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 (7),
$ k $ is the number of successes (1),
$ p $ is the probability of success (0.2).

Step 1: Identify the Parameters

We are given a binomial distribution problem where:

The number of trials, $ n = 7 $.
The number of successes, $ k = 1 $.
The probability of success, $ p = 0.2 $.

Step 2: Apply the Binomial Probability Formula

The probability of exactly $ k $ successes in $ n $ trials is given by the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Substituting the given values: \[ P(X = 1) = \binom{7}{1} (0.2)^1 (1-0.2)^{7-1} \]

Step 3: Calculate the Binomial Coefficient

The binomial coefficient $\binom{7}{1}$ is calculated as: \[ \binom{7}{1} = 7 \]

Step 4: Calculate the Probability

Substitute the values into the formula: \[ P(X = 1) = 7 \times 0.2 \times (0.8)^6 \] Calculate $(0.8)^6$: \[ (0.8)^6 = 0.262144 \] Thus, the probability is: \[ P(X = 1) = 7 \times 0.2 \times 0.262144 = 0.3670016 \]