Math Problem Statement

Question content area top

Part 1

​Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 16 consumers. Find the probability that the number of consumers who say it is important that the clothing they buy is made without child labor is​ (a) exactly seven ​(b) at least eleven​,

and​ (c) less than twelve.

Solution

This is a problem about binomial probability, as it involves a fixed number of independent trials (16 consumers), each of which results in one of two outcomes (either the consumer says it's important or they don't).

The key components for binomial probability are:

  • nn: number of trials (in this case, 16 consumers),
  • pp: probability of success (here, 45% or 0.45),
  • q=1pq = 1 - p: probability of failure (here, 10.45=0.551 - 0.45 = 0.55),
  • XX: the number of successes (consumers who say it's important).

The binomial probability formula is:

P(X=k)=(nk)pkqnkP(X = k) = \binom{n}{k} \cdot p^k \cdot q^{n-k}

Where:

  • (nk)\binom{n}{k} is the binomial coefficient, calculated as n!k!(nk)!\frac{n!}{k!(n-k)!},
  • pkp^k is the probability of exactly kk successes,
  • qnkq^{n-k} is the probability of the remaining trials being failures.

Let's solve the problem step by step.

Part (a) - Find the probability that exactly 7 consumers say it is important

We need to calculate P(X=7)P(X = 7).

Using n=16n = 16, p=0.45p = 0.45, and q=0.55q = 0.55:

P(X=7)=(167)(0.45)7(0.55)9P(X = 7) = \binom{16}{7} \cdot (0.45)^7 \cdot (0.55)^9

Part (b) - Find the probability that at least 11 consumers say it is important

We need to calculate P(X11)P(X \geq 11), which is the sum of the probabilities from 11 to 16.

P(X11)=P(X=11)+P(X=12)+P(X=13)+P(X=14)+P(X=15)+P(X=16)P(X \geq 11) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16)

Part (c) - Find the probability that fewer than 12 consumers say it is important

We need to calculate P(X<12)P(X < 12), which is the sum of the probabilities from 0 to 11.

P(X<12)=P(X=0)+P(X=1)++P(X=11)P(X < 12) = P(X = 0) + P(X = 1) + \cdots + P(X = 11)

I will now calculate these values for you.Here are the results for the given problem:

(a) The probability that exactly 7 consumers say it is important is:

P(X=7)0.1969P(X = 7) \approx 0.1969

(b) The probability that at least 11 consumers say it is important is:

P(X11)0.0486P(X \geq 11) \approx 0.0486

(c) The probability that fewer than 12 consumers say it is important is:

P(X<12)0.9851P(X < 12) \approx 0.9851

Would you like further details or have any questions? Here are some related questions to explore further:

  1. How does the probability change if the number of consumers increases to 20?
  2. What happens if the probability that a consumer says it's important changes to 50%?
  3. How is the binomial distribution connected to the normal distribution?
  4. What is the mean and variance of this binomial distribution?
  5. How would the probabilities change if you were sampling with a different probability (e.g., 30%)?

Tip: The sum of all probabilities in a binomial distribution always equals 1.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Probability
Binomial Distribution

Formulas

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
P(X ≥ k) = Σ P(X = i) for i = k to n
P(X < k) = Σ P(X = i) for i = 0 to k - 1

Theorems

Binomial Theorem

Suitable Grade Level

Grades 10-12