Probability and conditional probability

What this SAT skill is testing

SAT probability questions test whether you can identify the correct group of outcomes and use the correct denominator. The most important distinction is between an ordinary probability and a conditional probability.

Core formulas

• (P(A)=rac{ ext{outcomes in }A}{ ext{total outcomes}})
• (P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B))
• For independent events, (P(A\text{ and }B)=P(A)P(B)).
• (P(A\mid B)=rac{P(A\text{ and }B)}{P(B)})

Conditional probability changes the denominator. The phrase “given that” tells you which restricted group becomes the new total.

Worked two-way-table example

A club has 80 members. Of the 50 juniors, 20 volunteer. Of the 30 seniors, 18 volunteer.

• The probability that a randomly selected member volunteers is (rac{20+18}{80}=rac{38}{80}).
• The probability that a member volunteers given that the member is a senior is (rac{18}{30}).
• The probability that a volunteer is a senior is (rac{18}{38}). Notice that reversing the condition changes the denominator.

Independent and mutually exclusive events

Independent events do not affect one another. Mutually exclusive events cannot happen together. These ideas are not the same. If two events are mutually exclusive and both are possible, knowing that one occurred changes the probability of the other to zero, so they are not independent.

Step-by-step method

1. Translate the event into plain language.
2. Circle any condition introduced by given, among, or of those who.
3. Use that conditioned group as the denominator.
4. Count the outcomes that satisfy the full event for the numerator.
5. Check that the probability is between 0 and 1.

Common traps

• Using the grand total for a conditional probability.
• Adding overlapping probabilities without subtracting the overlap.
• Multiplying probabilities without confirming independence.
• Reversing (P(A\mid B)) and (P(B\mid A)).
• Reporting a count when the question asks for a probability or percent.

Practice strategy

On every table question, physically mark the denominator group first. Once the correct group is boxed, the numerator is usually easy to identify.