A binomial experiment with probability of success and trials is conducted. What is the probability that the experiment results in fewer 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 fewer 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 fewer 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 is the sum of the probabilities of getting 0 successes and 1 success:

$$
P(X < 2) = P(X = 0) + P(X = 1)
$$

Now, we will calculate $$ P(X = 0) $$ and $$ P(X = 1) $$:

1. **Calculate $$ P(X = 0) $$**:
   $$
   P(X = 0) = \binom{7}{0} (0.12)^0 (0.88)^7
   $$

2. **Calculate $$ P(X = 1) $$**:
   $$
   P(X = 1) = \binom{7}{1} (0.12)^1 (0.88)^6
   $$

Now, let's compute these probabilities step by step.

First, we need to calculate the binomial coefficients and the powers of $$ p $$ and $$ (1-p) $$.

Let's compute the values for $$ P(X = 0) $$ and $$ P(X = 1) $$.

The binomial coefficients are:
- $$ \binom{7}{0} = 1 $$
- $$ \binom{7}{1} = 7 $$

Now we can calculate the probabilities.

1. For $$ P(X = 0) $$:
   $$
   P(X = 0) = 1 \cdot (0.12)^0 \cdot (0.88)^7
   $$

2. For $$ P(X = 1) $$:
   $$
   P(X = 1) = 7 \cdot (0.12)^1 \cdot (0.88)^6
   $$

Now, I will calculate these values.
Calculate the value by following steps:
- step0: Calculate:
  $$0.88^{7}$$
- step1: Convert the expressions:
  $$\left(\frac{22}{25}\right)^{7}$$
- step2: Use the properties of exponents:
  $$\frac{22^{7}}{25^{7}}$$
Calculate or simplify the expression $$ 7 * (0.12) * (0.88)^6 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$7\times 0.12\times 0.88^{6}$$
- step1: Convert the expressions:
  $$7\times 0.12\left(\frac{22}{25}\right)^{6}$$
- step2: Multiply the terms:
  $$0.84\left(\frac{22}{25}\right)^{6}$$
- step3: Convert the expressions:
  $$\frac{21}{25}\left(\frac{22}{25}\right)^{6}$$
- step4: Multiply the numbers:
  $$\frac{21\times 22^{6}}{25^{7}}$$
Now we have the calculations for both probabilities:

1. For $$ P(X = 0) $$:
   $$
   P(X = 0) \approx 0.408676
   $$

2. For $$ P(X = 1) $$:
   $$
   P(X = 1) \approx 0.390099
   $$

Now, we can find $$ P(X < 2) $$ by summing these two probabilities:

$$
P(X < 2) = P(X = 0) + P(X = 1) \approx 0.408676 + 0.390099
$$

Let's calculate this sum.
Calculate the value by following steps:
- step0: Calculate:
  $$0.408676+0.390099$$
- step1: Add the numbers:
  $$0.798775$$
The total probability of getting fewer than 2 successes is:

$$
P(X < 2) \approx 0.798775
$$

Rounding this to three decimal places, we get:

$$
P(X < 2) \approx 0.799
$$

Thus, the probability that the experiment results in fewer than 2 successes is approximately **0.799**.
The probability of fewer than 2 successes is approximately 0.799.

A binomial experiment with probability of success and trials is conducted. What is the probability that the experiment results in fewer 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

Mind Expander

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A binomial experiment with probability of success and trials is conducted. What is the probability that the experiment results in fewer 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

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