Introduction to the Binomial Distribution - AP Statistics

1. What is $P(X=n)$ for $X \sim \text{Bin}(n,p)$?

   $p^n$. All trials succeed when each has probability $p$.

2. Which option correctly interprets $P(X=0)$ for $X\sim\text{Bin}(n,p)$?

   No successes in $n$ trials: $P(X=0)=(1-p)^n$. All failures occur with probability $(1-p)^n$.

3. What is the formula for the binomial coefficient $\binom{n}{k}$?

   $\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Counts ways to choose $k$ items from $n$.

4. Find $\mu$ and $\sigma$ for $X\sim\text{Bin}(25,0.4)$.

   $\mu=10,\ \sigma=\sqrt{6}$. Use $\mu=np=25(0.4)$ and $\sigma=\sqrt{25(0.4)(0.6)}$.

5. Find $P(X\ge 1)$ for $X\sim\text{Bin}(6,p)$ in terms of $p$.

   $1-(1-p)^6$. Use complement: $P(X\ge 1)=1-P(X=0)$.

6. What is the mean of a binomial random variable $X\sim\text{Bin}(n,p)$?

   $\mu=np$. Expected number of successes in $n$ trials.

7. Identify whether this is binomial: sample 5 cards without replacement, $X$ = number of aces.

   No; trials are not independent (without replacement). Removing cards changes probabilities between draws.

8. Which option lists the four binomial conditions as $BINS$?

   $BINS$: Binary outcomes, Independent trials, fixed Number of trials, Same $p$. Mnemonic for Binary, Independent, Number fixed, Same probability.

9. What four conditions must hold for a setting to be binomial (BINS)?

   Binary outcomes, Independent trials, fixed Number $n$, Same $p$ each trial. These ensure each trial has two outcomes with constant probability.

10. Find $P(X\le 1)$ for $X \sim \text{Bin}(4,0.3)$ as a sum of exact probabilities.

    $\binom{4}{0}(0.3)^0(0.7)^4+\binom{4}{1}(0.3)^1(0.7)^3$. Sum probabilities for $X=0$ and $X=1$.

11. Identify whether this is binomial: 10 coin flips, let $X$ be number of heads.

    Yes; $X \sim \text{Bin}(10,0.5)$. Fixed trials, constant $p=0.5$, independent flips.

12. Identify whether this is binomial: sample $20$ students without replacement from a class of $20$ and count left-handed.

    No; trials are not independent (without replacement from the entire class). Sampling entire population changes probabilities.

13. Identify the missing binomial condition: outcomes are success or failure, $n$ fixed, $p$ constant, and ____.

    Trials are independent. Independence ensures constant $p$ across all trials.

14. State the probability formula for exactly $k$ successes when $X\sim\text{Bin}(n,p)$.

    $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Combines choosing positions with probability of outcomes.

15. What is the standard deviation of $X\sim\text{Bin}(n,p)$?

    $\sigma=\sqrt{np(1-p)}$. Measures typical deviation from the mean.

16. Which option describes the key difference between geometric and binomial settings?

    Binomial: fixed $n$; geometric: trials continue until first success. Binomial has predetermined trials; geometric stops at success.

17. Find $P(X\ge 2)$ for $X \sim \text{Bin}(6,0.1)$ using a complement.

    $1-[P(X=0)+P(X=1)]$. Use complement: $P(X≥2)=1-P(X≤1)$.

18. What does the parameter $p$ represent in a binomial model $X\sim\text{Bin}(n,p)$?

    The probability of success on each trial. Remains constant across all trials in the experiment.

19. Which option gives the range of possible values for $X \sim \text{Bin}(n,p)$?

    $X \in \{0,1,2,\dots,n\}$. Can have 0 to $n$ successes in $n$ trials.

20. What is $P(X=0)$ for $X \sim \text{Bin}(n,p)$?

    $(1-p)^n$. All trials fail when each has probability $(1-p)$.

21. What is the support (possible values) of $X$ when $X\sim\text{Bin}(n,p)$?

    Integers $0,1,2,\dots,n$. Can have anywhere from zero to all successes.

22. Identify the distribution of the number of failures $Y=n-X$ when $X\sim\text{Bin}(n,p)$.

    $Y\sim\text{Bin}(n,1-p)$. Failures have probability $1-p$ each.

23. Find $P(X=3)$ for $X\sim\text{Bin}(5,0.2)$.

    $\binom{5}{3}(0.2)^3(0.8)^2$. Apply formula with $n=5$, $k=3$, $p=0.2$.

24. What is the mean of $X \sim \text{Bin}(n,p)$?

    $\mu=np$. Expected successes equals trials times success probability.

25. What is the variance of $X \sim \text{Bin}(n,p)$?

    $\text{Var}(X)=np(1-p)$. Product of mean $np$ and failure probability $(1-p)$.

26. What does the parameter $n$ represent in a binomial model $X\sim\text{Bin}(n,p)$?

    The fixed number of trials. How many times the experiment is repeated.

27. What is the variance of $X\sim\text{Bin}(n,p)$?

    $\text{Var}(X)=np(1-p)$. Square of standard deviation; measures spread.

28. Find $\mu$ and $\sigma$ for $X \sim \text{Bin}(20,0.4)$.

    $\mu=8,\ \sigma=\sqrt{4.8}$. $\mu=20(0.4)=8$; $\sigma=\sqrt{20(0.4)(0.6)}=\sqrt{4.8}$.

29. State the probability formula for exactly $k$ successes: $P(X=k)$ for $X \sim \text{Bin}(n,p)$.

    $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$. Combines choosing $k$ spots with probability of exact outcome.

30. State the formula for combinations $\binom{n}{k}$ using factorials.

    $\binom{n}{k}=\frac{n!}{k!(n-k)!}$. Counts arrangements divided by duplicate orderings.

31. Find $P(X\le 1)$ for $X\sim\text{Bin}(4,0.3)$ using binomial probabilities.

    $\binom{4}{0}(0.7)^4+\binom{4}{1}(0.3)(0.7)^3$. Sum probabilities for $X=0$ and $X=1$.

32. What is the binomial distribution used to model in AP Statistics?

    The number of successes $X$ in $n$ independent trials with constant $p$. Counts how many times an event occurs in repeated independent trials.

33. Identify whether this is binomial: flip a fair coin $10$ times and count heads.

    Yes; $X\sim\text{Bin}(10,0.5)$. Each flip is independent with $p=0.5$.

34. What is the standard deviation of $X \sim \text{Bin}(n,p)$?

    $\sigma=\sqrt{np(1-p)}$. Square root of variance $np(1-p)$.

35. Identify the complement event to "at least $k$ successes" for a binomial $X$.

    "At most $k-1$ successes," i.e., $X\le k-1$. "At least $k$" means $X\ge k$; complement is $X<k$.

36. Identify the distribution of the indicator variable $I$ for one trial with success probability $p$.

    $I \sim \text{Bernoulli}(p)$. Single trial with two outcomes is Bernoulli.

37. What is the meaning of the notation $\binom{n}{k}$ in the binomial probability formula?

    The number of ways to choose $k$ successes from $n$ trials. Also called "n choose k" or combinations.

38. What is the notation and meaning of $X \sim \text{Bin}(n,p)$?

    $X$ counts successes in $n$ trials with success probability $p$. Standard notation for binomial random variable.

39. Find $P(X=3)$ for $X \sim \text{Bin}(5,0.2)$.

    $\binom{5}{3}(0.2)^3(0.8)^2$. Apply formula with $n=5$, $k=3$, $p=0.2$.

Study these flashcards