Statistics & Probability

How to do it:

1. Name the population you want to talk about.

2. Give every member a fair chance to be picked (a random sample).

3. Check the question: no guilt-trip wording, and a sample large enough that a second survey would look similar.

Worked example: Standing only at the snack line to ask about Spirit Week wristbands overweights hungry shoppers. Drawing 40 names from shuffled advisory rosters is closer to random.

Check: bias is not "wrong arithmetic." It is a mismatch between the people you measured and the population you advertise. A cafeteria Instagram poll is not a school-wide random sample.

How to do it:

1. Expect two random samples from the same hallway to give slightly different means. That wiggle is sampling variability.

2. When you compare two groups, plot both distributions.

3. Look at center, spread, and overlap before you declare a winner.

Worked example: Club A homework times sit near 40 minutes with a tight cluster. Club B has the same mean but stretches from 10 to 90. The means match; the stories do not.

Check: if the two clouds sit on top of each other, a small gap in means is not strong evidence. Widen the sample before you lock a budget.

How to do it:

1. Start with a uniform model: fair die, equal spinner sectors, each outcome $\tfrac{1}{n}\text{.}$

2. For compound events, list outcomes with a table or tree.

3. Ask whether stages are independent (with replacement) or dependent (without).

Worked example: A fair eight-slice spinner labeled $1$–$8\text{.}$ $P(\text{even})=\tfrac{4}{8}=\tfrac{1}{2}\text{.}$ Two raffle tickets drawn without replacement are dependent: the second probability shifts.

Check: do not multiply as if tickets went back in the hat. Independent stages can multiply; dependent stages cannot use the same $P$ twice.

How to do it:

1. Estimate, then simulate: guess how often doubles appear, then tally 50 rolls.

2. Swap survey questions with a classmate and hunt for biased wording.

3. Match two dot plots to two different real-world captions before you compute.

Worked example: Guess $P(\text{doubles})$ on two dice as $\tfrac{1}{6}\text{,}$ then tally. The long-run fraction should sit near $\tfrac{1}{6}\text{,}$ not your first ten rolls.

Check: streaming "effects," sports streaks, and weather "40 percent rain" all need this toolkit. A loud sample is still a sample.