SAT Probability and Conditional Probability: The Complete Guide

The complement rule is $P(\text{not A})=1-P(A)$. It is useful when the event is easier to exclude than count directly. For an inclusive “or,” use $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$. Subtracting the overlap repairs the double counting caused by adding both totals.

Independence means learning that one event occurred does not change the probability of the other. In a table, compare the conditional proportion with the overall proportion. Do not infer independence merely because the categories have different names. A table can reveal a relationship between grade level and participation even though they describe different attributes.

More generally, $P(A\text{ and }B)=P(B)P(A\mid B)$ when $P(B)>0$. For 100 visitors, suppose 40 are members and 18 of those members attend a talk. Then $P(B)=40/100$ for membership and $P(A\mid B)=18/40$ for talk attendance given membership. Their product is $18/100$, the probability of both. The middle count cancels, showing why the conditional factor works without requiring independence. Only if talk attendance has the same probability among members as overall can you replace the conditional factor with the overall attendance probability.

For sequential selections, check replacement. Replacing an object restores the original composition before the next draw. Without replacement, both the total and potentially the favorable count change. You can multiply probabilities along a specific sequence, but each later factor must describe the eligible group at that stage. Independence should be established, not assumed.

A table can contain counts, percentages, or relative frequencies. Read the headings before adding cells. If percentages are computed separately within each row, they may sum to 100% in every row; adding them down a column does not generally produce a meaningful whole-table count or percent. You need row sizes to combine row percentages with proper weights.

Use a quick size check: probability must lie between 0 and 1 inclusive. A numerator larger than the denominator means your eligible group or count is wrong. An answer inside that range is not proof of correctness, but an answer outside it is an immediate signal to revisit the setup.

When a total row and total column are displayed, do not add them to the interior cells. They summarize the same people, so doing so counts observations repeatedly. Choose one way to get the overall count: sum the interior cells once, sum the non-total row totals, or read the bottom-right overall total.

A restriction can involve an exclusion. If selection is from students who are not in orchestra, use the not-orchestra column as the denominator. Do not subtract a probability computed over everyone unless you have kept the same eligible group. Complements work within a fixed sample space; they do not erase conditions.

Mutually exclusive events cannot occur together in one outcome. They are different from independent events. For a single token draw, red and blue are mutually exclusive: knowing the token is red makes blue impossible. They are therefore not independent when both colors have positive probability. In a two-way table, attributes such as grade level and workshop choice can overlap, so inspect the cells instead of assuming the categories exclude one another.

A missing cell can be found from its row total or column total. If a row total is 52 and one of its two exclusive cells is 19, the other is 33. Recover counts before computing probabilities. If a table's totals are inconsistent, reread whether an entry is a count or a percentage rather than forcing the numbers to fit.

For two draws without replacement from 5 blue and 3 yellow tokens, the probability both are blue is $\frac{5}{8}\times\frac{4}{7}=\frac{5}{14}$. After one blue token is removed, four blue remain among seven tokens. Using $5/8$ twice would describe replacement, a different experiment.

To find at least one success, consider the complement no successes. For two draws without replacement, calculate the chance of two failures using the changing counts, then subtract from 1. This avoids separately counting exactly one success in two orders and exactly two successes. It is a counting shortcut, not an assumption of independence.

If all you know are two overall probabilities, their overlap may not be determined. For example, knowing the share who bike and the share who swim does not identify how many do both. A question asking for an exact intersection needs additional information or an explicit independence assumption. Do not invent one because multiplication looks convenient.

Spend about 20 minutes on a focused session. For the first five minutes, use one two-way table and write four different fraction setups: one unconditional event, an intersection, and both directions of a conditional probability. Do not calculate until the denominator sentence is clear. This makes the effect of the wording visible.

Then solve Solar probability questions for ten minutes. Include a complement, an inclusive union, and a restricted table problem. For each answer, write favorable eligible members over all eligible members. If you miss a question, identify whether the error was in the numerator, the denominator, or the selection process.

In the last five minutes, rewrite a missed conditional question with its condition reversed and solve the new version. Return later for a mixed percentage-and-probability set. Under time pressure, keep the denominator label even if you stop writing full sentences; it is the smallest step with the greatest protection against a plausible distractor.