# SOCR EduMaterials Activities Binomial Distributions

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## This is an activity to explore the Binomial, Geometric, and Hypergeometric Probability Distributions.

• Exercise 1: Use SOCR to graph and print the following distributions and answer the questions below. Also, comment on the shape of each one of these distributions:
• a. $X \sim b(10,0.5)$, find P(X = 3), E(X), sd(X), and verify them with the formulas discussed in class.
• b. $X \sim b(10,0.1)$, find $P(1 \le X \le 3)$.
• c. $X \sim b(10,0.9)$, find $P(5 < X < 8), \ P(X < 8), \ P(X \le 7), \ P(X \ge 9)$.
• d. $X \sim b(30,0.1)$, find P(X > 2).

Below you can see a snapshot of the distribution of $X \sim b(20,0.3)$ • Exercise 2: Use SOCR to graph and print the distribution of a geometric random variable with p = 0.2,p = 0.7. What is the shape of these distributions? What happens when p is large? What happens when p is small?

Below you can see a snapshot of the distribution of $X \sim geometric(0.4)$ • Exercise 3: Select the geometric probability distribution with p = 0.2. Use SOCR to compute the following:
• a. P(X = 5)
• b. P(X > 3)
• c. $P(X \le 5)$
• d. P(X > 6)
• e. $P(X \ge 8)$
• f. $P(4 \le X \le 9)$
• g. P(4 < X < 9)
• Exercise 4: Verify that your answers in exercise 3 agree with the formulas discussed in class, for example, P(X = x) = (1 − p)x − 1p, P(X > k) = (1 − p)k, etc. Write all your answers in detail using those formulas.
• Exercise 5: Let X follow the hypergeometric probability distribution with N = 52, n = 10, and number of "hot" items 13. Use SOCR to graph and print this distribution.

Below you can see a snapshot of the distribution of $X \sim hypergeometric(N=100, n=15, r=30)$ • Exercise 6: Refer to exercise 5. Use SOCR to compute P(X = 5) and write down the formula that gives this answer.
• Exercise 7: Binomial approximation to hypergeometric: Let X follow the hypergeometric probability distribution with $N=1000, \ n=10$ and number of "hot" items 50. Graph and print this distribution.
• Exercise 8: Refer to exercise 7. Use SOCR to compute the exact probability: P(X = 2). Approximate P(X = 2) using the binomial distribution. Is the approximation good? Why?
• Exercise 9: Do you think you can approximate well the hypergeometric probability distribution with $N=50, \ n=20$, and number of "hot" items 40 using the binomial probability distribution? Graph and print the exact (hypergeometric) and the approximate (binomial) distributions and compare.

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