# SOCR EduMaterials Activities Binomial Distributions

## 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, $$P(X=x)=(1-p)^{x-1}p$$, $$P(X > k)=(1-p)^k$$, 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.