A binomial experiment with probability of success and trials is conducted. What is the probability that the experiment results in more than 2
successes?
Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

Question

A binomial experiment with probability of success

and

trials is conducted. What is the probability that the experiment results in more than 2
successes?
Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

UpStudy AI · Accepted Answer

To find the probability of getting more than 2 successes in a binomial experiment, we can use the binomial probability formula:

$$
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
$$

where:
- $$ n $$ is the number of trials,
- $$ k $$ is the number of successes,
- $$ p $$ is the probability of success,
- $$ \binom{n}{k} $$ is the binomial coefficient.

In this case, we want to find $$ P(X > 2) $$, which can be calculated as:

$$
P(X > 2) = 1 - P(X \leq 2) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))
$$

We will calculate $$ P(X = 0) $$, $$ P(X = 1) $$, and $$ P(X = 2) $$ using the given values $$ p = 0.37 $$ and $$ n = 9 $$.

Let's calculate each of these probabilities step by step:

1. **Calculate $$ P(X = 0) $$**:
   $$
   P(X = 0) = \binom{9}{0} (0.37)^0 (0.63)^9
   $$

2. **Calculate $$ P(X = 1) $$**:
   $$
   P(X = 1) = \binom{9}{1} (0.37)^1 (0.63)^8
   $$

3. **Calculate $$ P(X = 2) $$**:
   $$
   P(X = 2) = \binom{9}{2} (0.37)^2 (0.63)^7
   $$

Now, I will compute these probabilities.
Simplify the expression by following steps:
- step0: Solution:
  $$binom\times 9\times 0.37^{0}\times 0.63^{9}$$
- step1: Evaluate the power:
  $$binom\times 9\times 1\times 0.63^{9}$$
- step2: Convert the expressions:
  $$binom\times 9\times 1\times \left(\frac{63}{100}\right)^{9}$$
- step3: Rewrite the expression:
  $$binom\times 9\left(\frac{63}{100}\right)^{9}$$
- step4: Reorder the terms:
  $$ibnom\times 9\left(\frac{63}{100}\right)^{9}$$
- step5: Multiply the numbers:
  $$9ibnom\left(\frac{63}{100}\right)^{9}$$
- step6: Multiply the numbers:
  $$9\times \frac{63^{9}}{100^{9}}ibnom$$
- step7: Multiply:
  $$\frac{9\times 63^{9}}{100^{9}}ibnom$$
Calculate or simplify the expression $$ binom(9,2) * (0.37)^2 * (0.63)^7 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$binom\times 9.2\times 0.37^{2}\times 0.63^{7}$$
- step1: Convert the expressions:
  $$binom\times 9.2\left(\frac{37}{100}\right)^{2}\times 0.63^{7}$$
- step2: Convert the expressions:
  $$binom\times 9.2\left(\frac{37}{100}\right)^{2}\left(\frac{63}{100}\right)^{7}$$
- step3: Reorder the terms:
  $$ibnom\times 9.2\left(\frac{37}{100}\right)^{2}\left(\frac{63}{100}\right)^{7}$$
- step4: Multiply the numbers:
  $$9.2ibnom\left(\frac{37}{100}\right)^{2}\left(\frac{63}{100}\right)^{7}$$
- step5: Multiply the numbers:
  $$\frac{31487}{25000}ibnom\left(\frac{63}{100}\right)^{7}$$
- step6: Multiply the numbers:
  $$\frac{31487\times 63^{7}}{25000\times 100^{7}}ibnom$$
Calculate or simplify the expression $$ binom(9,1) * (0.37)^1 * (0.63)^8 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$binom\times 9.1\times 0.37^{1}\times 0.63^{8}$$
- step1: Calculate:
  $$binom\times 9.1\times \frac{37}{100}\times 0.63^{8}$$
- step2: Convert the expressions:
  $$binom\times 9.1\times \frac{37}{100}\left(\frac{63}{100}\right)^{8}$$
- step3: Multiply the terms:
  $$binom\times \frac{3367}{1000}\left(\frac{63}{100}\right)^{8}$$
- step4: Reorder the terms:
  $$ibnom\times \frac{3367}{1000}\left(\frac{63}{100}\right)^{8}$$
- step5: Multiply the numbers:
  $$\frac{3367}{1000}ibnom\left(\frac{63}{100}\right)^{8}$$
- step6: Multiply the numbers:
  $$\frac{3367\times 63^{8}}{10^{19}}ibnom$$
It seems there was an issue with the calculations. Let's compute the probabilities $$ P(X = 0) $$, $$ P(X = 1) $$, and $$ P(X = 2) $$ again using the binomial coefficients and the probabilities directly.

1. **Calculate $$ P(X = 0) $$**:
   $$
   P(X = 0) = \binom{9}{0} (0.37)^0 (0.63)^9 = 1 \cdot 1 \cdot (0.63)^9
   $$

2. **Calculate $$ P(X = 1) $$**:
   $$
   P(X = 1) = \binom{9}{1} (0.37)^1 (0.63)^8 = 9 \cdot (0.37) \cdot (0.63)^8
   $$

3. **Calculate $$ P(X = 2) $$**:
   $$
   P(X = 2) = \binom{9}{2} (0.37)^2 (0.63)^7 = 36 \cdot (0.37)^2 \cdot (0.63)^7
   $$

Now, I will compute these probabilities step by step.
Calculate the value by following steps:
- step0: Calculate:
  $$9\times 0.37\times 0.63^{8}$$
- step1: Convert the expressions:
  $$9\times 0.37\left(\frac{63}{100}\right)^{8}$$
- step2: Multiply the terms:
  $$3.33\left(\frac{63}{100}\right)^{8}$$
- step3: Convert the expressions:
  $$\frac{333}{100}\left(\frac{63}{100}\right)^{8}$$
- step4: Multiply the numbers:
  $$\frac{333\times 63^{8}}{100^{9}}$$
Calculate or simplify the expression $$ (0.63)^9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.63^{9}$$
- step1: Convert the expressions:
  $$\left(\frac{63}{100}\right)^{9}$$
- step2: Use the properties of exponents:
  $$\frac{63^{9}}{100^{9}}$$
Calculate or simplify the expression $$ 36 * (0.37)^2 * (0.63)^7 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$36\times 0.37^{2}\times 0.63^{7}$$
- step1: Convert the expressions:
  $$36\left(\frac{37}{100}\right)^{2}\times 0.63^{7}$$
- step2: Convert the expressions:
  $$36\left(\frac{37}{100}\right)^{2}\left(\frac{63}{100}\right)^{7}$$
- step3: Multiply the terms:
  $$\frac{12321}{2500}\left(\frac{63}{100}\right)^{7}$$
- step4: Evaluate the power:
  $$\frac{12321}{2500}\times \frac{63^{7}}{100^{7}}$$
- step5: Multiply the fractions:
  $$\frac{12321\times 63^{7}}{2500\times 100^{7}}$$
The calculated probabilities are as follows:

1. **For $$ P(X = 0) $$**:
   $$
   P(X = 0) \approx 0.015634
   $$

2. **For $$ P(X = 1) $$**:
   $$
   P(X = 1) \approx 0.082636
   $$

3. **For $$ P(X = 2) $$**:
   $$
   P(X = 2) \approx 0.194129
   $$

Now, we can find $$ P(X \leq 2) $$:
$$
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
$$

Calculating this sum:
$$
P(X \leq 2) \approx 0.015634 + 0.082636 + 0.194129
$$

Let's compute this total.
Calculate the value by following steps:
- step0: Calculate:
  $$0.015634+0.082636+0.194129$$
- step1: Add the numbers:
  $$0.292399$$
The total probability $$ P(X \leq 2) $$ is approximately:

$$
P(X \leq 2) \approx 0.292399
$$

Now, we can find $$ P(X > 2) $$:

$$
P(X > 2) = 1 - P(X \leq 2) \approx 1 - 0.292399
$$

Let's compute this final probability.
Calculate the value by following steps:
- step0: Calculate:
  $$1-0.292399$$
- step1: Subtract the numbers:
  $$0.707601$$
The probability of getting more than 2 successes in the binomial experiment is approximately:

$$
P(X > 2) \approx 0.707601
$$

Rounding this to three decimal places, we get:

$$
P(X > 2) \approx 0.708
$$

Thus, the final answer is $$ 0.708 $$.
The probability of getting more than 2 successes is approximately 0.708.

A binomial experiment with probability of success and trials is conducted. What is the probability that the experiment results in more than 2
successes?
Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

Upstudy AI Solution

Answer

Solution

The Deep Dive

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A binomial experiment with probability of success and trials is conducted. What is the probability that the experiment results in more than 2 successes? Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)

Upstudy AI Solution

Answer

Solution

The Deep Dive

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A binomial experiment with probability of success and trials is conducted. What is the probability that the experiment results in more than 2
successes?
Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.)