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Question 960871: When Teen Talk Barbie uttered those immortal words “Math class is tough!” in 1992 the American Association of University Women criticized this so strongly that Mattel deleted the phrase from the doll’s vocabulary and offered a swap to anyone who had such a doll. Each doll was programmed to say 4 different phrases randomly selected from a list of 270 phrases such as “Will we ever have enough clothes?”, “I love shopping!”, and “Wanna have a pizza party?” What percentage of the dolls manufactured at that time uttered the offending phrase about math class?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! presumably the phrase was part of the original vocabulary of 270 phrases.
each doll was programmed to use 4 of the 270 phrases.
the probability that one of the phrases would be the offending phrase is 1/270.
the assumption is that each doll could only get one of the offending phrase because each doll had to get 4 different phrases.
the doll either got the offending phrase or it didn't.
the probability is the number of ways the doll could get the offending phrase divided by the number of ways the doll could get 4 phrases regardless if the offending phrase was one of them or not.
using combination formulas, this would be equal to [ c(1,1) * c(269,3) ] divided by c(270,4).
this is equal to (1 * 3208094) / 216546345 = .0148148148
alternatively, using probability formulas, this can be viewed as:
1/270 * 4 = 4/270 = .0148148148.
the first method uses the combination formula of the number of ways you can get the offending phrase divided by the number of ways you can get any 4 phrases.
the number of ways you can get the offending phrase is 1 out of 1 multiplied by the number of ways you can get 3 of the remaining 269.
that's the c(1,1) * (c(269,3) part of the formula.
the number of ways you can get any 4 phrases is equal to the number of ways you can get 4 out of all 270.
the whole formula becomes (c(1,1) * (c(269,3)) / c(270,4)
c(n,x) is the combination formula of n! / (x! * (n-x)!).
the second methods uses the probability of getting the 1 offending phrase multiplied by the number of ways it could happen.
on the first pick, the probability is 1/270 * 269/269 * 268/268 * 267/267 which is equal to 1/270.
the same probability can happen on the second pick.
that probability would be 269/270 * 1/269 * 268/268 * 267*267.
the same probability can happen on the third pick.
that probability would be 269/270 * 268/269 * 1/268 * 267 * 267.
etc.
since you can pick the offending phrase on the first or second or third or fourth pick, the probability is 4 * 1/270 = 4/270.
2/270 is the same as (1 * 3208094) / 216546345
they are both equal to .0148148148.
same answer both ways says the answer is probably good, assuming you didn't mess up on both formulas.
best way to determine if you did it right is to take a much simpler example and work it through.
i did with 3 out of 5 rather than 4 out of 270.
the test indicated my original formulas were wrong.
i then corrected my mistake and that's what you see above.
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