Binomial and other Distributions

Instructor Katherine Cliff

Warm up

Classify each situation as geometric, binomial, or negative binomial. Do not calculate the probability.

A very good darts player can hit the bull’s eye (red circle in the center of the dart board) 65% of the time. What is the probability that she hits the bullseye 10 times in 15 tries?

For a sociology class project you are asked to conduct a survey on 20 students at your school.

You decide to stand outside of your dorm’s cafeteria and conduct the survey on a random sample of 20 students leaving the cafeteria after dinner one evening. Your dorm is comprised of 45% males and 55% females. What is the probability that the 4th person you survey is the 2nd female?

A not-so-skilled volleyball player has a 15% chance of making the serve,

which involves hitting the ball so it passes over the net on a trajectory such that it will land in the opposing team’s court. Suppose that her serves are independent of each other. What is the probability that on her 10th try she makes her first successful serve?

Example 1

We flip a coin five times. What is the probability that we get exactly 2 heads?

Example 1

We flip a coin five times. What is the probability that we see the first head on the fifth flip?

Example 1

We flip a coin five times. What is the probability that we see the third head on the fifth flip?

Example 2

We flip a coin 300 times. This is an unfair coin, with a \(\frac23\) probability of landing on tails. What is the probability that we get exactly 100 heads?

Example 2

We flip a coin 300 times. This is an unfair coin, with a \(\frac23\) probability of landing on tails. What is the probability that we see more than 100 heads?

Example 2

We flip a coin 300 times. This is an unfair coin, with a \(\frac23\) probability of landing on tails. What is the probability that we see between 50 and 100 heads (inclusive)?

Definition 1: The normal approximation to the binomial distribution

Continuity corrections

Example 3

You have a really annoying stapler that seems to randomly jam. On any given attempt to staple, it seems to independently jam 11.8% of the time.

Out of 280 papers stapled, what is the probability that your stapler will jam 26 times or fewer?

Example 3

You have a really annoying stapler that seems to randomly jam. On any given attempt to staple, it seems to independently jam 11.8% of the time.

Find the mean and standard deviation for the number of jammed staples out of 280 attempts.

Example 3

You have a really annoying stapler that seems to randomly jam. On any given attempt to staple, it seems to independently jam 11.8% of the time.

Does the normal approximation apply? Explain.

Example 3

You have a really annoying stapler that seems to randomly jam. On any given attempt to staple, it seems to independently jam 11.8% of the time.

Use the normal distribution (without continuity correction) to find the probability that your stapler will jam 26 times or fewer out of 280 attempts.

Example 3

You have a really annoying stapler that seems to randomly jam. On any given attempt to staple, it seems to independently jam 11.8% of the time.

Use the normal distribution (with continuity correction) to find the probability that your stapler will jam 26 times or fewer out of 280 attempts.

Work on the rest of the questions in your groups.