Probability
Simple and compound probability, independent and dependent events, expected values
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Have you ever reached into a giant bowl of Halloween candy blindfolded and hoped to pull out a chocolate bar instead of a boring mint? 🍫 That feeling of hoping for a specific outcome is exactly what probability is all about! Probability is just the math way of asking, "How likely is this to happen?"
To find the probability of something, you just need to know two things: what you want to happen (like getting chocolate), and everything that could possibly happen (every single piece of candy in the bowl). You just put what you want on top of what is possible to make a fraction! 🍕 For example, if a pizza has 8 slices and 2 of them have pepperoni, your chance of grabbing a pepperoni slice in the dark is .
On the SSAT, you will see a lot of questions about bags of colorful marbles, spinning game wheels, or drawing cards. Sometimes they ask about doing something once (simple probability), and sometimes they ask about doing it twice in a row (compound probability). Don't worry, the math is exactly the same! You just have to read carefully to see if you put the marble back in the bag before drawing again. You've got this! 🎲
Practice Questions
4 practice questions for SSAT Upper Level
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- First, find the total number of marbles in the jar: 40 + 30 + 15 + 15 = 100 marbles. Next, find the number of marbles that are either blue or green: 30 blue + 15 green = 45 marbles. The probability of selecting a blue or green marble is the number of blue and green marbles divided by the total number of marbles: . Simplify this fraction by dividing the numerator and denominator by 5 to get .
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- First, determine the total number of integers in the interval . The integers are 11, 12, 13, ..., 30. There are 30 - 11 + 1 = 20 integers in total. Next, identify the integers in this range that are divisible by 2 (even numbers): 12, 14, 16, 18, 20, 22, 24, 26, 28, 30. There are 10 such numbers. Now, eliminate the numbers that are also divisible by 4: 12, 16, 20, 24, and 28. The remaining numbers are divisible by 2 but not by 4: 14, 18, 22, 26, and 30. There are 5 such numbers. The probability is , which simplifies to .
A number is to be selected at random from the list above. What is the probability that the number selected will be a multiple of 3?
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- Count the total number of items in the list. There are 10 numbers. Next, identify which of the numbers are multiples of 3. The multiples of 3 in the list are 3, 3, 6, 9, 12, and 15. There are 6 multiples of 3 in total. The probability of selecting a multiple of 3 is therefore . Simplifying this fraction by dividing the numerator and denominator by 2 gives .
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- First, find the total number of socks in the drawer: 8 + 6 + 4 = 18 socks. The question asks for the probability that the selected sock is NOT black, which means it must be either blue or white. The number of blue and white socks is 6 + 4 = 10. The probability is the number of non-black socks divided by the total number of socks, which is . Dividing the numerator and denominator by 2 simplifies this to .
Tips & Strategies
- Always circle the words 'with replacement' or 'without replacement' in the question. If it says 'without', remember to subtract 1 from the top AND bottom of your fraction for the second draw!
- Simplify your fractions as early as possible. Multiplying is much easier than multiplying without a calculator.
- If a question asks for the probability of 'at least one' of something happening, find the probability of it NEVER happening, and subtract that from 1. It saves so much time!
Common Mistakes
- Watch out for forgetting to change the total number of items! If you eat a piece of candy from a bag, there is one less piece of candy overall. Your denominator MUST change for the next draw.
- Don't forget that when you want two things to happen in a row (like drawing a red marble AND a blue marble), you must MULTIPLY the fractions, not add them.
Frequently Asked Questions
What does 'without replacement' mean?
It means once you take an item out (like a marble or a card), you keep it out! You do not put it back before drawing the next one. This changes the total number of items left.
How do I know whether to add or multiply fractions in probability?
If you want event A AND event B to happen (like drawing two red marbles in a row), you multiply. If you want event A OR event B to happen (like drawing a red marble or a blue marble on a single try), you add.
Is probability always a fraction?
Usually! It is most often written as a fraction like , but it can also be written as a decimal (0.25) or a percentage (25%).
Will I get to use a calculator for these big fractions on the SSAT?
Nope, there are no calculators allowed on the SSAT. But don't panic! The test makers design the numbers so they will simplify nicely if you look for common factors.