Expert: Vivian - 5/26/2008

Question
"Synopsis: A survey found that 23% of respondents stick with their tour group.  
(No total number of travelers was given).



In a sample of 6 travelers, what is the probability that two will stick with
their group?



In a sample of 6 travelers, what is the probability that AT LEAST two will stick
with their group?



In a sample of 10 travelers, what is the probability that NONE will stick with
their group?



Any help on determining the answers will be very appreciated!  Thanks in
advance!"  

Answer
The total number of travelers who stick with their groups out of a sample has the binomial distribution with the probability that a traveler stick with his group, 0.23.

Probability that two out of 6 travelers will stick with their group is the binomial coefficient ‘6 choose 2’* 0.23^2*(1-0.23) ^4=0.28. You may obtain it by BINOMDIST(2,6,0.23,FALSE) in microsoft excel.

Probability that at least two out of 6 travelers will stick with their group is 1- [the binomial coefficient ‘6 choose 1’* 0.23^1*(1-0.23) ^5 + the binomial coefficient ‘6 choose 0’* 0.23^0*(1-0.23) ^6 ]=0.42. You may obtain it by 1-BINOMDIST(1,6,0.23,TRUE) in microsoft excel.

Probability that at least NONE out of 10 travelers will stick with their group is the binomial coefficient ‘10 choose 0’* 0.23^0*(1-0.23) ^10=0.07. You may obtain it by BINOMDIST(0,10,0.23,FALSE) in microsoft excel.