Parameters for a Binomial Distribution - AP Statistics

1. What is a common criterion for using normal approximation in a binomial distribution?

   $np \geq 10$ and $n(1-p) \geq 10$. Rule of thumb for normal approximation.

2. What distribution is used when modeling the total number of successes in $n$ trials?

   Binomial distribution. Models count of successes in fixed trials.

3. In a binomial distribution, what happens as $n$ increases with fixed $p$?

   Distribution becomes more symmetric. Central Limit Theorem effect.

4. State the condition for a binomial setting regarding trials.

   Trials must be independent. Outcome of one trial doesn't affect others.

5. What is the range of possible values for $p$ in a binomial distribution?

   $0 \leq p \leq 1$. Probability must be between 0 and 1 inclusive.

6. Choose the word that completes: A binomial distribution models the number of _____ in n trials.

   successes. Counts favorable outcomes in fixed trials.

7. If $p=0.5$ and $n=8$, find the probability of exactly 4 successes.

   $P(X=4) = 0.2734$. Using $\binom{8}{4}(0.5)^8 = 70 \times \frac{1}{256}$.

8. State the condition for a binomial setting regarding outcomes.

   Each trial has two possible outcomes: success or failure. Binary outcomes define binomial setting.

9. If $n=5$ and $p=0.2$, what is the probability of exactly 1 success?

   $P(X=1) = 0.4096$. Using $\binom{5}{1}(0.2)(0.8)^4 = 5 \times 0.2 \times 0.4096$.

10. Identify the distribution when $n=1$ in a binomial distribution.

    Bernoulli distribution. Single trial binomial distribution.

11. Calculate $P(X=0)$ for $n=3$ and $p=0.7$ using the binomial probability formula.

    $P(X=0) = 0.027$. Using $(1-0.7)^3 = (0.3)^3 = 0.027$.

12. What does the parameter $n$ represent in a binomial distribution?

    The number of trials. Fixed number of independent trials.

13. What is the binomial coefficient in the binomial probability formula?

    $\binom{n}{k}$. "Choose $k" from $n$ trials.

14. What is the binomial probability formula?

    $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. Combination term times probability terms.

15. What is the probability of no successes in any binomial distribution?

    $(1-p)^n$. All trials fail with probability $(1-p)$.

16. Calculate the variance if $n=20$ and $p=0.3$ for a binomial distribution.

    $Var(X) = 4.2$. Using $Var(X) = np(1-p) = 20 \times 0.3 \times 0.7 = 4.2$.

17. Find the probability of exactly 3 successes in 4 trials with $p=0.5$.

    $P(X=3) = 0.25$. Using $\binom{4}{3}(0.5)^3(0.5)^1 = 4 \times 0.0625 = 0.25$.

18. What does the parameter $p$ represent in a binomial distribution?

    The probability of success. Constant probability for each trial.

19. Determine the mean number of successes for $n=12$ and $p=0.25$.

    $E(X) = 3$. Using $E(X) = np = 12 \times 0.25 = 3$.

20. State the condition for a binomial setting regarding probability.

    Probability of success is constant for each trial. Same probability for every trial.

21. Identify the type of random variable a binomial distribution models.

    Discrete random variable. Counts whole numbers of successes only.

22. What is the shape of a binomial distribution when $p=0.5$?

    Symmetrical. Equal probability of success and failure creates symmetry.

23. What happens to the distribution shape as $p$ approaches 0 or 1?

    It becomes skewed. Extreme values create asymmetry.

24. Calculate the standard deviation if $n=15$ and $p=0.6$.

    $SD(X) = 1.89$. Using $SD(X) = \sqrt{15 \times 0.6 \times 0.4} = \sqrt{3.6} \approx 1.89$.

25. What is the formula for the expected value of a binomial distribution?

    $E(X) = n \times p$. Mean of binomial distribution equals trials times probability.

26. Find the expected number of successes if $n=10$ and $p=0.5$.

    $E(X) = 5$. Using $E(X) = np = 10 \times 0.5 = 5$.

27. What is the relationship between a binomial distribution and a normal distribution?

    Normal approximation when $n$ is large and $p$ is not near 0 or 1. Normal approximation works for large $n$.

28. What is the formula for the variance of a binomial distribution?

    $Var(X) = n \times p \times (1-p)$. Variance formula includes the complement $(1-p)$ term.

29. What is required for trials to be considered independent in a binomial setting?

    Outcome of one trial does not affect another. No influence between separate trials.

30. For $n=6$ and $p=0.3$, what is the probability of 0 successes?

    $P(X=0) = 0.1176$. Using $(1-0.3)^6 = (0.7)^6 = 0.1176$.

31. What is the nature of individual events in a binomial distribution?

    Mutually exclusive. Cannot occur simultaneously.

32. What is the formula for the standard deviation of a binomial distribution?

    $SD(X) = \sqrt{n \times p \times (1-p)}$. Standard deviation is the square root of variance.

33. Identify the two parameters of a binomial distribution.

    $n$ (number of trials) and $p$ (probability of success). These define the distribution completely.

34. How are binomial probabilities calculated?

    Using the binomial probability formula. Uses combination and probability of success/failure.

35. If $n=5$ and $p=0.4$, calculate $P(X=2)$. Use binomial probability formula.

    $P(X=2) = 0.3456$. Using $\binom{5}{2}(0.4)^2(0.6)^3 = 10 \times 0.16 \times 0.216$.

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