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ycow [4]
4 years ago
14

Sixty percent of the eligible voting residents of a certain community support the incumbent candidate. If eight of the residents

are selected at random, find the probability that at least five of the eight support the candidate.
Mathematics
1 answer:
il63 [147K]4 years ago
6 0

By the binomial theorem we know that

1 = (.4 + .6)^8 \\ = {8 \choose 0} (.4)^{8} (.6)^{0} + {8 \choose 1} (.4)^{7} (.6)^{1} +{8 \choose 2} (.4)^{6} (.6)^{2} + {8 \choose 3} (.4)^{5} (.6)^{3} + {8 \choose 4} (.4)^{4} (.6)^{4} \\ + \quad {8 \choose 5} (.4)^{3} (.6)^{5} + {8 \choose 6} (.4)^{2} (.6)^{6} + {8 \choose 7} (.4)^{1} (.6)^{7} + {8 \choose 8} (.4)^{0} (.6)^{8}

The probability that exactly 5 of 8 support the incumbent is the term

 {8 \choose 5} (.4)^{3} (.6)^{5}

So at least five of eight support is the sum of this term and beyond,

p={8 \choose 5} (.4)^{3} (.6)^{5} + {8 \choose 6} (.4)^{2} (.6)^{6} + {8 \choose 7} (.4)^{1} (.6)^{7} + {8 \choose 8} (.4)^{0} (.6)^{8}

No particularly easy way of calculating that except popping it into Wolfram Alpha which reports

p = \dfrac{ 46413}{78125}

Shouldn't half the terms work out to .6 ?  Interestingly it's not exactly .6 but pretty close at .594.

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Given that:

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The hull speed of the Santa Monica in terms of the length, ln of the Nina Pinta is given as a√b where a = 1.34 and b = ln - 10

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