Internet anywhere

In school, 60% of pupils have access to the internet at home. A group of 8 students is chosen at random. Find the probability that
a) exactly 5 have access to the internet.
b) At least 6 students have access to the internet

Correct answer:

a =  0.2787
b =  0.3154

Step-by-step explanation:

$$\begin{gathered} C_5(8)=\left({\displaystyle \genfrac{}{}{0px}{}{8}{5}}\right)={\displaystyle \frac{8!}{5!(8-5)!}}={\displaystyle \frac{8\cdot 7\cdot 6}{3\cdot 2\cdot 1}}=56 \\ p=60\mathrm{/}100={\displaystyle \frac{3}{5}}=0.6 \\ q=1-p=1-0.6={\displaystyle \frac{2}{5}}=0.4 \\ a=\left(\genfrac{}{}{0px}{}{5}{8}\right)\cdot p^5\cdot q^{8-5}=\left(\genfrac{}{}{0px}{}{5}{8}\right)\cdot 0.6^5\cdot 0.4^{8-5}=0.2787 \end{gathered}$$

$$\begin{gathered} C_6(8)=\left({\displaystyle \genfrac{}{}{0px}{}{8}{6}}\right)={\displaystyle \frac{8!}{6!(8-6)!}}={\displaystyle \frac{8\cdot 7}{2\cdot 1}}=28 \\ {C}_7(8)=\left({\displaystyle \genfrac{}{}{0px}{}{8}{7}}\right)={\displaystyle \frac{8!}{7!(8-7)!}}={\displaystyle \frac{8}{1}}=8 \\ {C}_8(8)=\left({\displaystyle \genfrac{}{}{0px}{}{8}{8}}\right)={\displaystyle \frac{8!}{8!(8-8)!}}={\displaystyle \frac{1}{1}}=1 \\ {p}_6=\left(\genfrac{}{}{0px}{}{6}{8}\right)\cdot p^6\cdot q^{8-6}=\left(\genfrac{}{}{0px}{}{6}{8}\right)\cdot 0.6^6\cdot 0.4^{8-6}\doteq 0.209 \\ {p}_7=\left(\genfrac{}{}{0px}{}{7}{8}\right)\cdot p^7\cdot q^{8-7}=\left(\genfrac{}{}{0px}{}{7}{8}\right)\cdot 0.6^7\cdot 0.4^{8-7}\doteq 0.0896 \\ {p}_8=\left(\genfrac{}{}{0px}{}{8}{8}\right)\cdot p^8\cdot q^{8-8}=1\cdot 0.6^8\cdot 0.4^{8-8}\doteq 0.0168 \\ b=p_6+p_7+p_8=0.209+0.0896+0.0168=0.3154 \end{gathered}$$

Try another example

Related math problems and questions:

• Family
  What is the probability that a family with 7 childrens have: exactly 6 dívek? 2 dívky and 5 boys? Consider the birth probability of a girl is 48.87% and boy 51.13%.
• Population variance
  In a California community college, 60% of students will transfer to a college in the CSU system. The number of students in a sample who will transfer follows a binomial distribution. If 8 students are randomly selected, find the population variance σ² of
• Covid-19
  Data showed that 22% of people in a small town was infected with the COVID-19 virus. A random sample of six residents from this town was selected. Find the probability that exactly two of these residents was infected.
• All use computer
  It is reported that 72% of working women use computers at work. Choose 3 women at random, find the probability that all 3 women use a computer in their jobs.
• Records
  Records indicate 90% error-free. If 8 records are randomly selected, what is the probability that at least 2 records have no errors?
• Doctors
  The drug successfully treats 90% of cases. Calculate the probability that he will cure at least 18 patients out of 20?
• Alopecia
  Medical literature indicates that 45% of men suffer from alopecia. For random sample of 8 men, calculate the probability that: (a) exactly four men suffer from alopecia. (b) at most two men suffer from alopecia.
• Binomial probability
  What is the binominal probability that at least 4 of the 6 trials (n=6) are succeeded where φ = 0.50?
• Seeds
  The germination of seeds of a certain species of carrot is 96%. What is the probability that at least 25 seeds out of 30 will germinate?
• Wimbledon finals
  Serena Williams made a successful first serve 67% 0f the time in a Wimbledon finals match against her sister Venus. If she continues to serve at the same rate the next time they play and serves 6 times in the first game, determine the probability that: 1.
• Mortality tables
  Mortality tables enable actuaries to obtain the probability that a person at any age will live a specified number of years. Insurance companies and others use such probabilities to determine life-insurance premiums, retirement pensions, and annuity paymen
• Dice
  We throw five times the dice. What is the probability that six fits exactly twice?
• The university
  At a certain university, 25% of students are in the business faculty. Of the students in the business faculty, 66% are males. However, only 52% of all students at the university are male. a. What is the probability that a student selected at random in the
• Level of significance
  At a certain college it is estimated that 25% of the students have cars on campus. Does this seem to be a valid estimate if in random sample of 90 college students, 28 are found to have cars? Use a 0.05 level of significance.
• Sales
  From statistics of sales goods, item A buy 51% of people and item B buys 59% of people. What is the probability that from 10 people buy 2 item A and 8 item B?
• Sick days
  In Canada, there are typically 261 working days per year. If there is a 4.9% chance that an employee takes a sick day. .. what is the probability an employee will use 17 OR MORE sick days in a year?
• Sample Proportion
  In a California community college, 60% of students will transfer to a college in the CSU system. The number of students in a sample who will transfer follows a binomial distribution. 400 students are randomly selected from the college and 224 of them will