Question

a weapons manufacturer uses a liquid propellant that can get mixed with another liquid to produce a contaminated cartridge. a statistician found that 24% of the cartridges in the particular lot were contaminated. suppose you randomly sample (without replacement) gun cartridges from this lot until you find a contaminated one. let x be the number of cartridges sampled until a contaminated one is found. it is known that the probability distribution for x is given by the formula shown below. complete parts a through c.
$p(x) = (0.24)(0.76)^{x - 1}, x = 1, 2, 3, \dots$

b. this value is the probability that one would first encounter a contaminated cartridge on the fourth trial.
c. this value is the probability that one would first encounter 4 contaminated cartridge in one hundred trials.

c. find $p(x \geq 2)$. interpret this result.
$p(x \geq 2) = \square$ (round to three decimal places as needed.)

Explanation:

Step1: Recall the complement rule

For any event \( A \), \( P(A) = 1 - P(\text{not } A) \). Here, \( P(x \geq 2)=1 - P(x = 1) \).

Step2: Calculate \( P(x = 1) \)

Using the formula \( p(x)=(0.24)(0.76)^{x - 1} \), when \( x = 1 \), we have \( p(1)=(0.24)(0.76)^{1 - 1}=(0.24)(0.76)^{0}=0.24\times1 = 0.24 \).

Step3: Calculate \( P(x \geq 2) \)

Using the complement rule, \( P(x \geq 2)=1 - P(x = 1)=1 - 0.24 = 0.76 \). Wait, no, wait. Wait, actually, the formula is for a geometric distribution. Wait, let's re - check. The geometric distribution gives the probability that the first success (finding a contaminated cartridge) occurs on the \( x \)-th trial. So \( P(x=k)=(1 - p)^{k - 1}p \), here \( p = 0.24 \), \( 1 - p=0.76 \).

To find \( P(x\geq2) \), we can also think of it as the probability that the first success is not on the first trial. So \( P(x\geq2)=1 - P(x = 1) \). But \( P(x = 1)=(0.24)(0.76)^{0}=0.24 \). So \( P(x\geq2)=1 - 0.24=0.76 \)? Wait, no, that seems wrong. Wait, no, wait, let's calculate \( P(x\geq2)=\sum_{x = 2}^{\infty}p(x) \). Since the sum of all probabilities \( \sum_{x = 1}^{\infty}p(x)=1 \), then \( \sum_{x = 2}^{\infty}p(x)=1-\sum_{x = 1}^{1}p(x)=1 - p(1) \). But \( p(1)=(0.24)(0.76)^{0}=0.24 \), so \( P(x\geq2)=1 - 0.24 = 0.76 \). Wait, but let's verify with the formula. Let's calculate \( P(x\geq2)=1 - P(x = 1)=1-(0.24)(0.76)^{0}=1 - 0.24 = 0.76 \). Yes, that's correct.

Answer:

\( 0.760 \)