From past experience, it is known 90% of one-year-old children can distinguish their mother's...


From past experience, it is known {eq}90\% {/eq} of one-year-old children can distinguish their mother's voice from the voice of a similar sounding female. A random sample of {eq}20 {/eq} one-year-old are given this voice recognition test. Find the probability at least 3 children do not recognize their mother's voice.

Binomial Probability Distribution:

Binomial distribution is a type of Bernoulli experiment with two possible outcomes termed as success or failure. If binomial trial is carried out once, it becomes a Bernoulli trial.

Answer and Explanation:

Given that 90% of the one year old children can recognize their mother's voice, 10% cannot recognize their mother's voice. If 20 one year old children are randomly selected, it becomes a binomial experiment with n=20 and p=0.1. Use binomial probability mass function to calculate the following probability:

{eq}\begin{align*} P(X\ge 3)&=1-\left[P(x=0)+P(x=1)+P(x=2)\right]\\b(n,x,p)&=^nC_xp^x(1-p)^{n-x}\\P(x=0)&\\b(20,0.1,0)&=^{20}C_0\cdot 0.1^0(1-0.1)^{20-0}\\&=1\times 1\times 0.9^{20}\\&=0.1216\\P(x=1)&\\b(20,0.1,1)&=^{20}C_1\cdot 0.1^1(1-0.1)^{20-1}\\&=0.2702\\P(x=2)&\\b(20,0.1,2)&=^{20}C_2\cdot 0.1^2(1-0.1)^{20-2}\\&=0.2852\\\therefore P(x\ge 3)&=1-(0.1216+0.2702+0.2852)\\&=0.323 \end{align*} {/eq}

Learn more about this topic:

Binomial Theorem Practice Problems

from Math 101: College Algebra

Chapter 11 / Lesson 4

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