Answer:
The probability that at least one of the 20 4th graders is addicted to comic books is 0.12.
Step-by-step explanation:
Let X = number of hours a 4th grader reads comics and Y = number of 4th grader addicted to reading comics.
The random variable X is continuous and follows a normal distribution with mean, μ = 6 hours and standard deviation, σ = 2 hours.
And the random variable Y is discrete and follows a binomial distribution with success defined as a 4th grader is addicted to reading comics.
Compute the probability that a 4th grader is addicted to reading comics as follows, i.e. determine the value of P (X > 11) .
![P(X>11) = P(\frac{X-\mu}{\sigma} >\frac{11-6}{2})\\=P(Z >2.5)\\=1-P(Z](https://tex.z-dn.net/?f=P%28X%3E11%29%20%3D%20P%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%20%3E%5Cfrac%7B11-6%7D%7B2%7D%29%5C%5C%3DP%28Z%20%3E2.5%29%5C%5C%3D1-P%28Z%3C2.5%29%5C%5C%3D1-0.9938%5C%5C%3D0.0062)
Use the <em>Z</em>-table for the probability value.
Now compute the probability that out of 20 fourth graders at least 1 is addicted to comics as follows:
The Binomial probability function is:
![P(Y)=^{n}C_{y} p^{y} (1-p)^{n-y}](https://tex.z-dn.net/?f=P%28Y%29%3D%5E%7Bn%7DC_%7By%7D%20p%5E%7By%7D%20%20%281-p%29%5E%7Bn-y%7D)
Compute the value of
as follows:
![P(Y\geq 1)=1-P(Y](https://tex.z-dn.net/?f=P%28Y%5Cgeq%201%29%3D1-P%28Y%3C1%29%5C%5C%3D1-P%28Y%3D0%29%5C%5C%3D1-%5B%5E%7B20%7DC_%7B20%7D%20%280.0062%29%5E%7B0%7D%20%20%281-0.0062%29%5E%7B20-0%7D%5C%5C%3D1-%281%5Ctimes1%5Ctimes0.88304%29%5C%5C%3D0.11696%5C%5C%5Capprox0.12)
Thus, the probability that at least one of the 20 4th graders is addicted to comic books is 0.12.