Question

There are 26 letters in the English alphabet. Letters are picked one by one at random, so that each letter has the same chance of appearing as any other letter, regardless of which letters have appeared or not appeared. a. Six letters are picked. Find the chance that the sequence that appears is RANDOM, in that order. b. Six letters are picked. Find the chance that they can be arranged to form the word RANDOM.

Answer 1

A person is selected at random from a population that has the following characteristics: 65% of the people are men; the others are women 12% of the men are smokers 7% of the women are smokers a. Find the chance that the selected person is a non-smoker, given that the person is a man. b. Find the chance that the selected person is a woman who smokes. c. Find the chance that the selected person is a smoker. d. Given that the selected person is a smoker, what is the chance that the person is a woman? e. Find the chance that the selected person is a man or a non-smoker.

Answer 2

it's relatively simple i): Note that the probability of picking the correct letter, in order, is 1/26, also note that RANDOMNESS is 6 letters long, hence:
P((A) of picking the correct letter, in order )=(1/26)^6

Where A is the event picking the exact letters, in order. Interestingly enough, there's not much need for the latter problem
ii)Note that the total possible number of rearrangements in RANDOMNESS is 6!=(6*5*4*3*2*2*1)=720, so, noting that we could get any of the one rearrangements from any of the 26^6 possible solutions, multiplying by the permutations gives us the probability of the favorable cases. Hence:
P((B) of rearrangements in RANDOMNESS is )=(( (1/26)^6)*6!= ((1/26)^6)*720