<h3>The Main Idea</h3> <p><strong>Mean is the balance point.</strong></p> <p>Think of the mean like balancing a pencil on your finger — the distances above and below the balance point must be equal. This understanding helps you solve problems forwards (calculate the mean) and backwards (find a missing value).</p> <h3>What are Mean, Median, and Mode?</h3> <p><strong>Mean</strong> (Average): Add up all the values and divide by how many there are. It's the "balance point" of the data.</p> <p><strong>Median</strong>: The middle value when the numbers are in order. If there's an even count, average the two middle numbers.</p> <p><strong>Mode</strong>: The value that appears most often. A data set can have no mode, one mode, or multiple modes.</p> <h3>Calculating Mean</h3> <ol> <li><strong>Add all the numbers together</strong> — Get the total sum.</li> <li><strong>Count how many numbers there are</strong> — This is your divisor.</li> <li><strong>Divide the sum by the count</strong> — That's your mean.</li> </ol> <h3>Calculating Median</h3> <ol> <li><strong>Order the numbers</strong> — Least to greatest (or greatest to least).</li> <li><strong>Find the middle</strong> — Cross off from both ends until you reach the center.</li> <li><strong>If odd count</strong> — The middle number is your median.</li> <li><strong>If even count</strong> — Average the two middle numbers.</li> </ol> <h3>Finding a Missing Value (Given the Mean)</h3> <p>Two approaches:</p> <p><strong>Method 1: Algebraic</strong></p> <ol> <li>Use the formula: Mean × Count = Total</li> <li>Calculate what the total must be</li> <li>Subtract the known values to find the missing one</li> </ol> <p><strong>Method 2: Balance Point</strong></p> <ol> <li>Find how far each known value is from the mean</li> <li>Values below the mean are "negative distance," above are "positive"</li> <li>The missing value must make the distances balance to zero</li> </ol> <h3>Common Pitfalls</h3> <ul> <li><strong>Forgetting to order for median</strong> — Always sort first!</li> <li><strong>Even vs. odd count</strong> — With even count, you must average the two middle values.</li> <li><strong>Confusing mean and median</strong> — Mean uses all values in a calculation; median just finds the middle position.</li> </ul>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Probability is favorable outcomes divided by total outcomes.</strong></p> <p>Every probability question comes down to: "How many ways can what I want happen?" divided by "How many things could happen total?"</p> <h3>Simple Probability</h3> <p>When one event happens once:</p> $$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$ <p>Probability is always between 0 (impossible) and 1 (certain).</p> <h3>Compound Probability: AND vs OR</h3> <p>When multiple events are involved, the key word tells you what to do:</p> <p><strong>"AND" = Multiply</strong></p> <ul> <li>Both events must happen</li> <li>P(A and B) = P(A) × P(B) — for independent events</li> <li>Think: The chance gets <em>smaller</em> because both must occur</li> </ul> <p><strong>"OR" = Add (then subtract overlap)</strong></p> <ul> <li>Either event can happen</li> <li>P(A or B) = P(A) + P(B) − P(A and B)</li> <li>Think: The chance gets <em>bigger</em> because either works</li> <li>If events can't happen together (mutually exclusive): just add, no subtraction needed</li> </ul> <h3>With vs Without Replacement</h3> <p><strong>With replacement:</strong> Put the item back before the next draw. Probabilities stay the same.</p> <p><strong>Without replacement:</strong> Don't put it back. Total outcomes decrease, and favorable may too.</p> <h3>Common Scenarios</h3> <ul> <li><strong>Coins:</strong> P(heads) = 1/2, P(tails) = 1/2</li> <li><strong>Dice:</strong> P(any single number) = 1/6, P(even) = 3/6 = 1/2</li> <li><strong>Cards:</strong> 52 total, 4 suits of 13, P(ace) = 4/52 = 1/13</li> <li><strong>Marbles/balls:</strong> Count colors, pay attention to replacement</li> </ul> <h3>The Complement Rule</h3> <p>Sometimes it's easier to find the probability of what you <em>don't</em> want:</p> $$P(\text{event}) = 1 - P(\text{not event})$$ <p>Example: P(at least one heads in 3 flips) = 1 − P(all tails) = 1 − (1/2)³ = 7/8</p> <h3>Common Pitfalls</h3> <ul> <li><strong>Confusing AND/OR:</strong> "And" shrinks probability (multiply), "or" grows it (add).</li> <li><strong>Forgetting replacement changes totals:</strong> Without replacement, denominator decreases each draw.</li> <li><strong>Double-counting with OR:</strong> If events overlap, subtract P(A and B) or you count it twice.</li> </ul>
Practice This Goal