Probability Basics: Any Event

Probability questions show up on the SHSAT and SAT almost every year—and they can feel tricky at first, especially when more than one event is involved. A common point of confusion is figuring out whether one event makes another event more (or less) likely.

Here’s the reassuring part: no matter how complicated a probability question looks, the final probability is always a number between 0 and 1.

• A probability of 0 means the event is impossible and will absolutely not happen.

• A probability of 1 means the event is guaranteed and will absolutely occur.

• Most answers fall somewhere in between (as a fraction, decimal, or percent--1 = 100%).

EXAMPLE (2018 SHSAT Test A, Question 78)

In a sample of 10 cards, 4 are red and 6 are blue. If 2 cards are selected at random from the sample, one at a time without replacement, what is the probability that both cards are not blue?

A) 2/15

B) 4/25

C) 3/10

D) 1/3

Solution: “Replacement” means the selected card is returned to the deck before the next draw. In this case, the question specifies :without replacement", so the first card is not returned.

Event 1: P(1st card is not blue) = 4/10 = 2/5

Event 2: P(2nd card is not blue, without replacement) = 3/9 = 1/3

Because we want Event 1 AND Event 2 to happen, we multiply the probabilities (we do not add them):

2/5 × 1/3 = 2/15

The ANSWER is choice A) 2/15.

Notice that the answer is a fraction between 0 and 1 and whose relatively small value tells us that it's an event whose occurrence would be somewhat uncommon in real life. Indeed, it's likelihood would about 2 times in 15 tries.