Question

An experiment succeeds twice as often as it fails. The chance that in the next six trials, there shall be at least four successes is

• A. ${}_4^{6}C{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^2$
• B. ${}_4^{6}C{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^2{+}_5^{6}C{\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^1$
• C. ${}_4^{6}C{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^2{+}_5^{6}C{\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^1+{\left(\frac{2}{3}\right)}^6$
• D. $1{-}_4^{6}C{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^2{-}_5^{6}C{\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^1-{\left(\frac{2}{3}\right)}^6$

Answer 1

Answer: (C)

${}_4^{6}C{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^2{+}_5^{6}C{\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^1+{\left(\frac{2}{3}\right)}^6$

Probability of an experiment which succeeds twice as often as it fails: $\frac{2}{3}$
Probability of its fail is: $\frac{1}{3}$
Selection of success four out of six event is: ${}^4{C}_3$
Probability for at least for success out of six events is $=P\left(4\right)+P\left(5\right)+P\left(6\right)$
${=}_4^{6}C{\left(\frac{2}{3}\right)}^4{\left(\frac{1}{3}\right)}^2{+}_5^{6}C{\left(\frac{2}{3}\right)}^5{\left(\frac{1}{3}\right)}^1+{\left(\frac{2}{3}\right)}^6$