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Strike441 [17]
3 years ago
14

A financial talk show host claims to have a 55.3 % success rate in his investment recommendations. You collect some data over th

e next few weeks, and find that out 10 recommendations, he was correct 3 times. If the claim is correct and the performance of recommendations is independent, what is the probability that you would have observed 4 or fewer successfu
Mathematics
1 answer:
baherus [9]3 years ago
3 0

Answer:

There is a 25.52% probability of observating 4 our fewer succesful recommendations.

Step-by-step explanation:

For each recommendation, there are only two possible outcomes. Either it was a success, or it was a failure. So we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

In which C_{n,x} is the number of different combinations of x objects from a set of n elements, given by the following formula.

C_{n,x} = \frac{n!}{x!(n-x)!}

And p is the probability of X happening.

In this problem we have that:

p = 0.553, n = 10

If the claim is correct and the performance of recommendations is independent, what is the probability that you would have observed 4 or fewer successful:

This is

P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

In which

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

P(X = 0) = C_{10,0}.(0.553)^{0}.(0.447)^{10} = 0.0003

P(X = 1) = C_{10,1}.(0.553)^{1}.(0.447)^{9} = 0.0039

P(X = 2) = C_{10,2}.(0.553)^{2}.(0.447)^{8} = 0.0219

P(X = 3) = C_{10,3}.(0.553)^{3}.(0.447)^{7} = 0.0724

P(X = 4) = C_{10,4}.(0.553)^{4}.(0.447)^{6} = 0.1567

P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.0003 + 0.0039 + 0.0219 + 0.0724 + 0.1567 = 0.2552

There is a 25.52% probability of observating 4 our fewer succesful recommendations.

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Answer:

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Step-by-step explanation:

8.96 ÷ 2.8= 3.2

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What is the true solution to the equation below?
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Answer:

The solution is:

  • x=4

Step-by-step explanation:

Considering the expression

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\ln \left(e^{\ln \left(x\right)}\right)=\ln \left(x\right),\:\space\ln \left(e^{\ln \left(x\right)2}\right)=\ln \left(x\right)2

\ln \left(x\right)+\ln \left(x\right)\cdot \:2=2\ln \left(8\right)

\mathrm{Add\:similar\:elements:}\:\ln \left(x\right)+2\ln \left(x\right)=3\ln \left(x\right)

3\ln \left(x\right)=2\ln \left(8\right)

\mathrm{Divide\:both\:sides\:by\:}3

\frac{3\ln \left(x\right)}{3}=\frac{2\ln \left(8\right)}{3}

\ln \left(x\right)=\frac{2\ln \left(8\right)}{3}.....A

Solving the right side of the equation A.

\frac{2\ln \left(8\right)}{3}

As

\ln \left(8\right):\quad 3\ln \left(2\right)

Because

\ln \left(8\right)

\mathrm{Rewrite\:}8\mathrm{\:in\:power-base\:form:}\quad 8=2^3

⇒ \ln \left(2^3\right)

\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)

\ln \left(2^3\right)=3\ln \left(2\right)

So

\frac{2\ln \left(8\right)}{3}=\frac{2\cdot \:3\ln \left(2\right)}{3}

\mathrm{Multiply\:the\:numbers:}\:2\cdot \:3=6

          =\frac{6\ln \left(2\right)}{3}

\mathrm{Divide\:the\:numbers:}\:\frac{6}{3}=2

          =2\ln \left(2\right)

So, equation A becomes

\ln \left(x\right)=2\ln \left(2\right)

\mathrm{Apply\:log\:rule}:\quad \:a\log _c\left(b\right)=\log _c\left(b^a\right)

         =\ln \left(2^2\right)

         =\ln \left(4\right)

\ln \left(x\right)=\ln \left(4\right)

\mathrm{Apply\:log\:rule:\:\:If}\:\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\:\mathrm{then}\:f\left(x\right)=g\left(x\right)          

x=4

Therefore, the solution is

  • x=4
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