Doctors

The drug successfully treats 90% of cases. Calculate the probability that he will cure at least 18 patients out of 20?

Correct result:

P =  0.6769 %

Solution:

$$\begin{gathered} p=90\mathrm{\%}={\displaystyle \frac{90}{100}}=0.9 \\ {C}_{18}(20)=\left({\displaystyle \genfrac{}{}{0px}{}{20}{18}}\right)={\displaystyle \frac{20!}{18!(20-18)!}}={\displaystyle \frac{20\cdot 19}{2\cdot 1}}=190 \\ {p}_{18}=\left(\genfrac{}{}{0px}{}{20}{18}\right)\cdot p^{18}\cdot (1-p{)}^2=190\cdot 0.9^{18}\cdot (1-0.9{)}^2\doteq 0.2852 \\ {C}_{19}(20)=\left({\displaystyle \genfrac{}{}{0px}{}{20}{19}}\right)={\displaystyle \frac{20!}{19!(20-19)!}}={\displaystyle \frac{20}{1}}=20 \\ {p}_{19}=\left(\genfrac{}{}{0px}{}{20}{19}\right)\cdot p^{19}\cdot (1-p{)}^1=20\cdot 0.9^{19}\cdot (1-0.9{)}^1\doteq 0.2702 \\ {C}_{20}(20)=\left({\displaystyle \genfrac{}{}{0px}{}{20}{20}}\right)={\displaystyle \frac{20!}{20!(20-20)!}}={\displaystyle \frac{1}{1}}=1 \\ {p}_{20}=\left(\genfrac{}{}{0px}{}{20}{20}\right)\cdot p^{20}\cdot (1-p{)}^0=1\cdot 0.9^{20}\cdot (1-0.9{)}^0\doteq 0.1216 \\ P=p_{18}+p_{19}+p_{20}=0.2852+0.2702+0.1216=0.6769\mathrm{\%} \end{gathered}$$

Try another example

Showing 0 comments:

Related math problems and questions:

• 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?
• Records
  Records indicate 90% error-free. If 8 records are randomly selected, what is the probability that at least 2 records have no errors?
• Medicine
  We test medicine on 6 patients. For all drug doesn't work. If the drug success rate of 20%, what is the probability that medicine does not work?
• Two doctors
  Doctor A will determine the correct diagnosis with a probability of 93% and doctor B with a probability of 79%. Calculate the probability of proper diagnosis if both doctors diagnose the patient.
• Probability of malaria
  A survey carried out at a certain hospital indicates that the probability that a patient tested positive for malaria is 0.6. What is the probability that two patients selected at random (i) one is negative while the other tested positive? (i) both patient
• Component fail
  There is a 90 percent chance that a particular type of component will perform adequately under high temperature conditions. If the device involved has four such components, determine the probability that the device is inoperable because exactly one of the
• Alarm systems
  What is the probability that at least one alarm system will signal the theft of a motor vehicle when the efficiency of the first system is 90% and of the independent second system 80%?
• Dice
  How many times must throw the dice, the probability of throwing at least one six was greater than 90%?
• Left handed
  It is known that 25% of the population is left-handed. What is the probability that there is a maximum of three left-handers at a seminar where there are 30 participants?
• Bernoulli distribution
  The production of solar cells produces 2% of defective cells. Assume the cells are independent and that a lot contains 800 cells. Approximate the probability that less than 20 cells are defective. (Answer to the nearest 3 decimals).
• 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
• Family
  94 boys are born per 100 girls. Determine the probability that there are two boys in a randomly selected family with three children.
• Boys and girls
  There are 11 boys and 18 girls in the classroom. Three pupils will answer. What is the probability that two boys will be among them?
• 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.
• 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?
• 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.
• 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.