SOLUTION: 14 The probability of winning a prize in game of chance is 0.2. The least number of games that must be played to ensure that the probability of winning at least one prize is more t

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Question 1043583: 14 The probability of winning a prize in game of chance is 0.2. The least number of games that must be played to ensure that the probability of winning at least one prize is more than 0.90 is closest to:
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The act of playing the game multiple times is a binomial experiment with p = 0.20.
The probability of winning at least one prize is given by the expression
sum%28%28matrix%282%2C1%2Cn%2Ck%29%29%2A0.8%5E%28n-k%29%2A0.2%5Ek%2C+k+=+1%2Cn%29.
Hence we want or
1+-+0.8%5En+%3E+0.9.
<===> 0.10+%3E+0.8%5En
===> ln%280.10%29+%3E+ln+%280.8%5En%29+=+n%2Aln%280.80%29, since ln is an increasing function.
===> ln%280.10%29%2Fln%280.80%29+=+10.31885116+%3C+n, since ln(0.80) < 0.
===> 11+%3C=+n.
Therefore least number of games that must be played to ensure that the probability of winning at least one prize is more than 0.90 is highlight%2811%29.