Question

question 4 of 5
thousands of travelers pass through the airport in guadalajara, mexico, each day. before leaving the airport, each passenger must go through the customs inspection area. customs agents want to be sure that passengers do not bring illegal items into the country, but they do not have time to search every traveler’s luggage. instead, they require each person to press a button. either a red or a green bulb lights up. if a red light flashes, the passenger will be searched by customs agents. a green light means it is ok for the passenger to “go ahead.” customs agents claim that the light has probability 0.30 of showing red on any push of the button. assume for now that this claim is true. suppose we watch 20 passengers press the button. let ( r = ) the number who get a red light.

(a) explain why ( r ) is a binomial random variable.
this is a binomial setting because:
binary?
\success\ = the light is red.
\failure\ = the light is not red.
independent?
knowing whether or not one passenger has a red light tells you nothing about whether or not another passenger gets a red light.
number? ( n = 20 )
same probability? ( p = 0.30 )

(b) find the probability that exactly 6 of the 20 passengers get a red light.
(round to 4 decimal places. leave your answer in decimal form.)

here is the histogram of the random variable ( r ).

Explanation:

Part (b) Solution:

Step1: Recall Binomial Probability Formula

The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \), where \( n \) is the number of trials, \( k \) is the number of successes, \( p \) is the probability of success on a single trial, and \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \).

Step2: Identify Values

Here, \( n = 20 \), \( k = 6 \), and \( p = 0.30 \). So \( 1 - p = 1 - 0.30 = 0.70 \).

Step3: Calculate Combination

First, calculate \( \binom{20}{6} \).
\[
\binom{20}{6} = \frac{20!}{6!(20 - 6)!} = \frac{20!}{6!14!} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 38760
\]

Step4: Calculate Probability

Now, substitute into the binomial formula:
\[
P(R = 6) = \binom{20}{6} (0.30)^6 (0.70)^{14}
\]
Calculate \( (0.30)^6 = 0.000729 \) and \( (0.70)^{14} \approx 0.067857 \).
Then, multiply these together with the combination:
\[
P(R = 6) = 38760 \times 0.000729 \times 0.067857 \approx 0.1916
\]

Answer:

\( 0.1916 \)