ISEE Upper 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 $0$ (no way, like a T-Rex showing up to take your test 🦖) to $1$ (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 $10$ gumballs and $3$ are your favorite blue raspberry flavor, your chance of getting a blue one is $3$ out of $10$ . On the ISEE, we write that as the fraction $\frac{3}{10}$ . 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 $6$ 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:

Probability = \frac{What you WANT}{Total possible outcomes}

Practice Questions

5 practice questions for ISEE Upper Level

Q1 Hard

A box contains 5 red tokens, 4 blue tokens, and 3 green tokens. Two tokens are drawn at random from the box without replacement.

Column A

The probability that both tokens drawn are red

Column B

The probability that both tokens drawn are blue

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

First, find the total number of tokens: $5 + 4 + 3 = 12$ . For Column A, the probability of drawing two red tokens without replacement is $\frac{5}{12} \times \frac{4}{11} = \frac{20}{132}$ . For Column B, the probability of drawing two blue tokens without replacement is $\frac{4}{12} \times \frac{3}{11} = \frac{12}{132}$ . Since $\frac{20}{132}$ is greater than $\frac{12}{132}$ , the quantity in Column A is greater.

Answer: A

Q2 Hard

A fair spinner is divided into 8 equal sectors: 3 are colored red, 4 are colored blue, and 1 is colored yellow. If the spinner is spun twice, what is the probability that the first spin lands on red and the second spin lands on yellow?

\frac{3}{64}

\frac{1}{16}

\frac{1}{8}

\frac{1}{2}

Show Solution

The probability of landing on red on the first spin is $\frac{3}{8}$ . The probability of landing on yellow on the second spin is $\frac{1}{8}$ . Because the spins are independent events, you multiply their probabilities together: $\frac{3}{8} \times \frac{1}{8} = \frac{3}{64}$ .

Answer: A

Q3 Hard

Two fair six-sided number cubes are rolled simultaneously. What is the probability that the sum of the numbers rolled is exactly 8?

\frac{1}{12}

\frac{5}{36}

\frac{1}{6}

\frac{7}{36}

Show Solution

There are $6 \times 6 = 36$ total possible outcomes when rolling two number cubes. The combinations that result in a sum of 8 are (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2). There are 5 successful outcomes out of 36 total possibilities. Therefore, the probability is $\frac{5}{36}$ .

Answer: B

Q4 Hard

A pencil case contains 4 green pens, 5 black pens, and 3 blue pens. If one pen is chosen at random and then returned to the case, and a second pen is chosen at random, what is the probability that both pens will be black?

\frac{5}{12}

\frac{25}{132}

\frac{5}{12} \times \frac{5}{12}

\frac{5}{12} \times \frac{4}{11}

Show Solution

There are $4 + 5 + 3 = 12$ total pens. The probability of choosing a black pen on the first draw is $\frac{5}{12}$ . Since the pen is returned to the case, the total number of pens and the number of black pens remain the same for the second draw. The probability of choosing a black pen on the second draw is also $\frac{5}{12}$ . To find the compound probability of these independent events, multiply them: $\frac{5}{12} \times \frac{5}{12}$ .

Answer: C

Q5 Hard

A fair coin is flipped exactly 3 times.

Column A

The probability of getting exactly 2 heads

Column B

The probability of getting exactly 1 head

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

There are $2^{3} = 8$ possible outcomes when flipping a coin 3 times. The outcomes with exactly 2 heads are HHT, HTH, and THH (3 outcomes), making the probability $\frac{3}{8}$ . The outcomes with exactly 1 head are HTT, THT, and TTH (3 outcomes), making the probability $\frac{3}{8}$ . Since $\frac{3}{8} = \frac{3}{8}$ , the quantities are equal.

Answer: C

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 $1$ (like $1 - \frac{1}{4} = \frac{3}{4}$ ).
Always simplify your fractions! The ISEE loves to hide the correct answer by reducing it. If you get $\frac{4}{8}$ , look for $\frac{1}{2}$ 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 $1$ .
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 $\frac{2}{6}$ , you should expect to see $\frac{1}{3}$ 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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