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Catracho Catracho
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5 years ago
Forty
Forty percent of households say they would feel secure if they had​ $50,000 in savings. You randomly select 8 households and ask them if they would feel secure if they had​ $50,000 in savings. Find the probability that the number that say they would feel secure is​ (a) exactly​ five, (b) more than​ five, and​ (c) at most five.

​(a) Find the probability that the number that say they would feel secure is exactly five.
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Valued Member
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5 years ago
For this you use the binomial distribution formula: \(P\left(r\right)=_nC_r\times \pi ^r\left(1-\pi \right)^{n-r}\), where \(\pi\) is the probability.

45% is equivalent to 0.45 in decimal form.

a) exactly​ five

For the probability of exactly 5, you set n = 8, r = 5, \(\pi\) = 0.45, and \(1-\pi =0.55\)

\(P\left(5\right)=_{8}C_5\times 0.45^3\left(0.55\right)^{8-5}\)

\(P\left(5\right)=_8C_5\times 0.45^3\left(0.55\right)^{8-5}=56\cdot 0.01516=0.849\)

b) more than​ five

For this, find the probabilities of 6, 7 and 8, and add them up.

\(P\left(r>5\right)=P\left(6\right)+P\left(7\right)+P\left(8\right)\)

\(_8C_6\times 0.45^6\left(0.55\right)^2+_8C_7\times 0.45^7\left(0.55\right)^1+_8C_8\times 0.45^8\left(0.55\right)^0=0.0884\rightarrow 0.088\)

c) at most five

1 minus the answer above. We already found the probabilities of greater than 5, to find at most five means 5 is the maximum.

\(P\left(r\le 5\right)=1-P\left(r>5\right)=1-0.08845=0.9115\rightarrow 0.912\)
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