Question

according to a recent survey, 80% of high - school students have their own cell phones. suppose you select 10 high - school students at random. determine each probability. round your answers to the nearest tenth of a percent if necessary.
sample problem
p(8 of the students have cell phones)
=₁₀c₈·(8/10)⁸×(2/10)²
≈45(0.0007)
≈0.3015
≈30.2%
p(all of the students have cell phones)
enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Identify the binomial - probability formula

The binomial - probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 10$ (number of students selected), $p = 0.8$ (probability that a student has a cell - phone), and $k$ is the number of students with cell - phones.

Step2: Calculate the combination for $k = 10$

$C(10,10)=\frac{10!}{10!(10 - 10)!}=\frac{10!}{10!0!}=1$.

Step3: Calculate the probability

$P(X = 10)=C(10,10)\times(0.8)^{10}\times(1 - 0.8)^{10 - 10}$. Substitute the values: $P(X = 10)=1\times(0.8)^{10}\times(0.2)^{0}$. Since any non - zero number to the power of 0 is 1, $(0.2)^{0}=1$. Then $(0.8)^{10}=0.1073741824$.

Step4: Convert to percentage and round

To convert to a percentage, multiply by 100: $0.1073741824\times100 = 10.73741824\%$. Rounding to the nearest tenth of a percent, we get $10.7\%$.

Answer:

$10.7\%$