Math Problem Statement

Determine the probability 

P2

 for a binomial experiment with 

=n5

 trials and success probability 

=p0.7

. Then find the mean, variance, and standard deviation.

Part: 0 / 3

0 of 3 Parts Complete

Part 1 of 3

Determine the probability 

P2

. Round the answer to at least four decimal places.

Solution

To determine the probability P(X=2)P(X = 2) for a binomial experiment, we use the binomial probability formula:

P(X=k)=(nk)pk(1p)nkP(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}

Where:

  • n=5n = 5 (the number of trials),
  • p=0.7p = 0.7 (the probability of success),
  • k=2k = 2 (the number of successes we want to calculate the probability for),
  • (nk)\binom{n}{k} is the binomial coefficient, calculated as:

(nk)=n!k!(nk)!\binom{n}{k} = \frac{n!}{k!(n-k)!}

Step 1: Compute the binomial coefficient (52)\binom{5}{2}:

(52)=5!2!(52)!=5×42×1=10\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10

Step 2: Apply the formula:

P(X=2)=10×(0.7)2×(10.7)52P(X = 2) = 10 \times (0.7)^2 \times (1 - 0.7)^{5-2}

P(X=2)=10×(0.49)×(0.3)3P(X = 2) = 10 \times (0.49) \times (0.3)^3

P(X=2)=10×0.49×0.027P(X = 2) = 10 \times 0.49 \times 0.027

P(X=2)=10×0.01323=0.1323P(X = 2) = 10 \times 0.01323 = 0.1323

Thus, the probability P(X=2)P(X = 2) is 0.1323 (rounded to four decimal places).


Part 2: Find the mean, variance, and standard deviation

For a binomial distribution, the following formulas apply:

  • Mean μ=n×p\mu = n \times p
  • Variance σ2=n×p×(1p)\sigma^2 = n \times p \times (1 - p)
  • Standard deviation σ=σ2\sigma = \sqrt{\sigma^2}

Step 1: Mean

μ=5×0.7=3.5\mu = 5 \times 0.7 = 3.5

Step 2: Variance

σ2=5×0.7×0.3=1.05\sigma^2 = 5 \times 0.7 \times 0.3 = 1.05

Step 3: Standard deviation

σ=1.051.0247\sigma = \sqrt{1.05} \approx 1.0247

Final results:

  • Mean = 3.5
  • Variance = 1.05
  • Standard deviation = 1.0247 (rounded to four decimal places)

Do you need further clarification or a more detailed explanation for any of the steps? Here are some related questions to explore further:

  1. What is the difference between a binomial and a normal distribution?
  2. How do you calculate the cumulative probability for a binomial distribution?
  3. How do you use the normal approximation to the binomial distribution?
  4. What happens to the binomial distribution as the number of trials increases?
  5. How do you find the probability of having more than 2 successes in a binomial distribution?

Tip: For larger values of nn, using normal approximation to the binomial distribution can be helpful, especially when calculating probabilities for a range of outcomes.

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Math Problem Analysis

Mathematical Concepts

Binomial Distribution
Probability
Statistics

Formulas

Binomial probability formula: P(X = k) = nCk * p^k * (1 - p)^(n-k)
Mean: μ = n * p
Variance: σ^2 = n * p * (1 - p)
Standard Deviation: σ = √σ^2

Theorems

Binomial Distribution Theorem

Suitable Grade Level

Grades 9-12