ISEE Middle Level

Probability

Simple and compound probability, independent and dependent events, expected values

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Have you ever tried to guess what’s for dinner before you smell it? If it’s Friday, there’s a really good chance it’s pizza! 🍕 That "chance" is what mathematicians call probability. Probability is just a fancy way of asking, "How likely is something to happen?"

It’s a huge part of the ISEE, showing up in both the Quantitative Reasoning and Mathematics Achievement sections. We measure probability on a scale from (no way, like a T-Rex showing up to take your test 🦖) to (definitely happening, like you doing an awesome job!). Most of the time, probability lives right in the middle as a fraction.

Think of a giant gumball machine. If there are gumballs and are your favorite blue raspberry flavor, your chance of getting a blue one is out of . On the ISEE, we write that as the fraction . The bottom number (denominator) is always the total number of things, and the top number (numerator) is the number of winning things!

Sometimes, the ISEE will ask you to figure out the chances of two things happening in a row. Like getting a blue gumball AND rolling a on a dice. Here is a secret test trick: when you want one thing AND another thing to happen, you multiply their probabilities together! 🎲 Get ready to be a probability pro!

Key Formula
To find the chance of something happening, use this rule:

Practice Questions

4 practice questions for ISEE Middle Level

Q1 Medium
A bag contains 5 red marbles, 4 blue marbles, and 3 yellow marbles. If one marble is selected at random, what is the probability that it will be blue?
A
B
C
D
Show Solution
  • There are marbles in total, but wait — . The probability of selecting a blue marble is .
Answer: B
Q2 Medium
A spinner is divided into 8 equal sections numbered 1 through 8. What is the probability of landing on a number greater than 5?
A
B
C
D
Show Solution
  • The numbers greater than 5 on the spinner are 6, 7, and 8 — that is 3 favorable outcomes. The total number of equally likely outcomes is 8. Therefore, the probability is .
Answer: B
Q3 Medium
A box contains 2 white, 3 gray, and 5 black socks. Sarah picks one sock at random. Which pair of events describes complementary events?
A Picking a white sock and picking a gray sock.
B Picking a black sock and picking a sock that is not black.
C Picking a white sock and picking a black sock.
D Picking a gray sock and picking a gray sock.
Show Solution
  • Complementary events are two outcomes that are mutually exclusive (cannot both happen) and collectively exhaustive (together they cover all possible outcomes), meaning their probabilities sum to 1. 'Picking a black sock' and 'picking a sock that is not black' together cover every possible outcome — you must pick one or the other — and they cannot happen simultaneously. The probability of black is and the probability of not black is also , which sum to 1.
Answer: B
Q4 Medium
Maria has 3 pairs of shorts, 5 shirts, and 2 pairs of sneakers. She randomly selects one item from each category to make an outfit. What is the probability that she picks her favorite pair of shorts, her favorite shirt, and her favorite pair of sneakers?
A
B
C
D
Show Solution
  • The probability of picking the favorite shorts is , the favorite shirt is , and the favorite sneakers is . Since these are independent choices, multiply: .
Answer: A

Tips & Strategies

  • The 'AND' vs 'OR' rule! On the ISEE, if you need Event A AND Event B to happen, multiply their fractions. If you need Event A OR Event B (and they can't both happen at the same time), add their fractions!
  • The 'NOT' trick! If a question asks for the probability of something NOT happening, find the probability of it happening, and subtract it from (like ).
  • Always simplify your fractions! The ISEE loves to hide the correct answer by reducing it. If you get , look for in the answer choices.

Common Mistakes

  • Watch out for 'without replacement'! 🛑 If a problem says someone takes a marble and keeps it, the total number of marbles for the next draw goes down by .
  • Don't forget to count all the possibilities in coin flips. Flipping Heads then Tails is a different outcome than flipping Tails then Heads!

Frequently Asked Questions

Do I need to simplify my fractions on the ISEE?

Yes! The ISEE almost always simplifies fractions in the answer choices. If you calculate , you should expect to see as the correct choice.

What does 'independent event' mean?

It means the first event doesn't change the chances of the second event! Like flipping a coin twice: the coin doesn't remember what it landed on the first time.

What is Quantitative Comparison?

It's a special question format on the ISEE Quantitative Reasoning section where you compare Column A and Column B. You just need to figure out which side is bigger, or if they are perfectly equal!

Is there a penalty for guessing on the ISEE?

Nope! You never lose points for a wrong answer on the ISEE. If you are stuck on a tough probability question, take your best guess and move on to the next one! ✨

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