A given binomial experiment has n=100 trials and p=1/3. Is it more likely to get x=20 successes or x=45 successes. Why?

Question

A given binomial experiment has n=100 trials and p=1/3. Is it more likely to get x=20 successes or x=45 successes. Why?

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Kaylee 2 weeks 2021-09-09T09:42:37+00:00 1 Answer 0

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    2021-09-09T09:44:07+00:00

    Answer:

    The P(x=45) is more that the P(x=20). Therefore x=45 successes is more likely to get.

    Step-by-step explanation:

    Given information: n=100 and p=1/3.

    According to the binomial distribution, the probability of getting r success in n trials is

    P(x=r)=^nC_rp^rq^{n-r}

    where, n is total trials, p is probability of success and q is probability of failure.

    Total trials, n = 100

    Probability of success, p = \frac{1}{3}

    Probability of failure, q = 1-\frac{1}{3}=\frac{2}{3}

    The probability of 20 successes is

    P(x=20)=^{100}C_{20}\times (\frac{1}{3})^{20}\times (\frac{2}{3})^{100-20}

    P(x=20)=\frac{100!}{20!(100-20)!}\times (\frac{1}{3})^{20}\times (\frac{2}{3})^{80}\approx 0.001257

    The probability of 45 successes is

    P(x=45)=^{100}C_{45}\times (\frac{1}{3})^{45}\times (\frac{2}{3})^{100-45}

    P(x=45)=\frac{100!}{45!(100-45)!}\times (\frac{1}{3})^{45}\times (\frac{2}{3})^{55}\approx 0.004296

    The P(x=45) is more that the P(x=20). Therefore x=45 successes is more likely to get.

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