Probability is about turning a story into a clean fraction: how many ways the event can happendivided by how many total ways are possible. Once you choose the right model (list outcomes, tree diagram, or counting), most problems become systematic.
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In many beginner probability problems, every outcome is equally likely (coins, fair dice, well-shuffled cards, random draws from a bag). In that case:
Your job is to correctly count both parts. The two most common mistakes are (1) counting the event wrong and (2) counting the total sample space wrong. Everything else is details.
“At least one” problems are often easiest using a complement:
If A and B cannot both happen (mutually exclusive):
If A and B are independent:
If they are not independent, use conditional probability.
The formal definition is:
If you want a clean reference for these rules (with examples like “at least one” and conditional probability), the OpenStax section on probability is a solid companion.
Many probability mistakes are really translation mistakes. Before you compute anything, rewrite the sentence using these “keyword” meanings.
Two phrases that often get confused: mutually exclusive (cannot both happen) is not the same as independent (one does not change the probability of the other). For example, “roll an even number” and “roll a 6” are not mutually exclusive (6 is even), but they also aren’t independent in the “cannot both happen” sense.
Problem: You roll a fair six-sided die. What is the probability of rolling a number greater than 4?
Outcomes are 1–6. “Greater than 4” means {5, 6}.
Quick check: rolling 5 or 6 is possible but not super common, so a probability around 0.33 makes sense.
Problem: You flip a fair coin 3 times. What is the probability of getting at least one head?
“At least one head” is easiest via the complement: “no heads,” meaning all tails.
Quick intuition: getting no heads in 3 flips is pretty unlikely, so 7/8 (0.875) makes sense.
Problem: A bag has 5 red and 3 blue marbles. You draw 2 marbles without replacement. What is the probability both are red?
Because you do not replace the first marble, the probability changes on the second draw.
Notice how “without replacement” is the clue that it’s conditional (dependent) probability.
Problem: A card is drawn from a standard deck. What is the probability the card is a heart or a king?
This is an OR problem. Hearts and kings overlap because the king of hearts is counted in both groups. Use the general addition rule:
The “− overlap” step is what prevents double counting. If your OR question involves two categories that can overlap, this rule is your best friend.
Sometimes listing outcomes is impossible because there are too many. That’s where counting methods help. The key question is: does order matter?
Example: choosing president and vice president (two different roles). The pair (A, B) is different from (B, A).
Example: choosing a 3-person committee. The set {A, B, C} is the same as {C, B, A}.
In probability, you often use counting to compute favorable outcomes and total outcomes. Then you still finish with favorable/total.
If a problem has multiple stages (two draws, three flips, two spins, “first this happens then that happens”), a tree diagram keeps you organized. Each branch represents a choice, and you multiply along a branch to get the probability of that path.
Quick rule: multiply along a branch (AND), and add across branches that match your event (OR).
Even if you don’t draw a perfect tree, sketching the first two levels often prevents mistakes like assuming independence when you should be using conditional probabilities.
Expected value shows up in games, lotteries, and “is this a good deal?” questions. It’s the long-run average result. If outcomes are x1, x2, ... with probabilities p1, p2, ... then:
Even if you do not compute EV often, knowing it helps you recognize when a probability question is really an EV question.
Tip: write a quick tree diagram for multi-step questions and circle the branches that match the event.
“And” usually means multiply (with conditional probability if needed). “Or” usually means add (but watch overlap).
“At least one” is often faster as 1 − P(none). It also reduces counting mistakes.
Without replacement, the second probability changes. That’s the #1 clue that events are dependent.
If your answer is greater than 1 or negative, something is wrong — re-check the sample space and arithmetic.
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