I write each of the letters ‘L’, ‘I’, ‘O’, ‘N’ on separate pieces of paper and put these into a hat. I close my eyes and begin picking the pieces of paper from the hat and reading off the letters written on the pieces of paper. a. If I pick 3 pieces of paper without replacement, then what is the probability that the letters can be rearranged to spell ‘OIL’? b. If I pick pieces of paper with replacement until the third ‘L’ appears, then what is the probability the third ‘L’ appears on the 5th selection?
a) Number of ways to select r items from n, nCr = n!/(r! x (n-r)!)
Number of letters in the hat = 4
Of the 4 letters, we need to pick the combination of O, I and L (in any order) so that we can rearrange the pieces to spell 'OIL'.
Number of ways to select any 3 letters from 4 = 4C3
= 4!/(3! x 1!)
P(the letters can be rearranged to spell ‘OIL’) = 1/Number of ways to select any 3 letters from 4
b) Since the selections are with replacement, the probability of selecting L follows Binomial distribution: P(X) = nCx px qn-x
P(selecting L), p = 1/4
P(not selecting L), q = 1 - 1/4 = 3/4
P(third 'L' appears on the 5th selection) = P(exactly 2 'L's on the first 4 selections and L on the fifth selection)
= 4C2 x (1/4)2 x (3/4)2 x (1/4)
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