Probability

TSI Math Study Guide

Probability

Probability measures how likely an event is to occur.

What’s tested: the probability of simple and compound events, the probability of a complement, distinguishing independent from dependent events, and interpreting probability in real-world situations.
0 — impossible0.51 — certain

0 ≤ P(event) ≤ 1

Probability of a simple event

When all possible outcomes are equally likely:

P(event) = number of favorable outcomestotal number of possible outcomes

Rolling a 5 on a number cube

123456

One favorable outcome out of six.P(5) = 16

Rolling an even number

123456

Three favorable outcomes out of six.P(even) = 36 = 12

Expressing probability

Probability can be written as a fraction, decimal, or percent. Being comfortable moving among the forms makes questions easier.

34 = 0.75 = 75%     0.2 = 20% = 15

Probability from data

Sometimes probability comes from observed data rather than a list of equally likely outcomes. The same relationship applies.

Transportation Students
Car 90
Bus 60
Walk 30
Bicycle 20
Problem200 students were surveyed. If one is selected at random, what is the probability that the student takes the bus?
P(bus) = 60200 = 310 = 30%

Complements

Complement — all the outcomes in which the event does not occur. An event and its complement together have probability 1.

P(A) + P(Ac) = 1,  so  P(Ac) = 1 − P(A)

ProblemThe probability that a flight arrives on time is 0.82. What is the probability that it does not?
P(not on time) = 1 − 0.82 = 0.18or 18%

Why it’s useful

The complement rule is often much faster than calculating every unwanted outcome separately.

Compound events

Compound event — involves two or more events. The words and and or usually indicate different operations.

“And” — usually multiply

ProblemA fair coin is flipped and a number cube is rolled. What is the probability of getting heads and rolling a 4?
P(heads) = 12    P(4) = 16 P(heads and 4) = 12 · 16 = 112multiply — the events are independent

Independent events

Independent — the outcome of one event doesn’t change the probability of the other.
flipping a coin twicerolling two separate number cubesrolling a cube and flipping a coin

If A and B are independent:  P(A and B) = P(A) · P(B)

Heads twice in a row P(H and H) = 12 · 12 = 14the first flip doesn’t affect the second

Dependent events

Dependent — the first event changes the probability of the second. This commonly happens when objects are selected without replacement.
ProblemA bag contains 5 red marbles and 3 blue marbles. Two marbles are selected without replacement. What is the probability that both are red?

First draw — 8 marbles, 5 red

P(first red) = 58

Second draw — 7 left, 4 red

P(second red) = 47

The denominator dropped from 8 to 7 because the first marble was not replaced.

P(both red) = 58 · 47 = 2056 = 514

Without replacement → dependent

58 · 47 = 514

With replacement → independent

58 · 58 = 2564

If the first marble goes back, the second draw is from the original 8.

⚠ Watch out

Always look for the phrases with replacement or without replacement.

“Or” — usually add

If the events can’t happen at the same time, simply add.

ProblemA number cube is rolled. What is the probability of rolling a 2 or a 5?
P(2 or 5) = 16 + 16 = 26 = 13one roll can’t be both 2 and 5

Overlapping events

If two events can happen at the same time, simply adding counts the overlap twice.

P(A or B) = P(A) + P(B) − P(A and B)

ProblemA number from 1 through 10 is selected at random. What is the probability that it is even or greater than 7?

Even

12345678910

Greater than 7

12345678910

8 and 10 belong to both groups. Counting each outcome once, the favorable outcomes are 2, 4, 6, 8, 9, 10.

P(even or > 7) = 610 = 356 favorable out of 10

Using two-way tables

Two-way tables organize information involving two categories.

Plays a sport Does not play Total
Grade 11 35 25 60
Grade 12 30 30 60
Total 65 55 120

Plays a sport

65120 = 1324

A column total.

Is in Grade 12

60120 = 12

A row total.

Grade 12 and plays a sport

30120 = 14

The intersection — the highlighted cell.

Read carefully

Does the question want a row total, a column total, or the intersection of two categories?

Interpreting probability in context

Probability describes what’s expected over many trials, not what must happen in a few. A basketball player who makes free throws with probability 0.80 isn’t guaranteed to make exactly 8 of the next 10. Over many attempts, the proportion would be expected to settle near 80%. Short-term results vary.

Expected number of outcomes

ProblemThe probability that a randomly selected customer makes a purchase is 0.30. If 500 customers visit, how many are expected to make a purchase?
500(0.30) = 150an expectation, not a guarantee

TSI strategy: look for “and,” “or,” and “not”

And

Usually multiply.P(A and B)

Or

Usually add — and subtract any overlap.P(A or B)

Not

Use the complement.1 − P(A)

Then check whether the events are independent or dependent. If objects are selected without replacement, the total changes after each selection.

🔑 Key tip

Before calculating a compound probability, ask: does the first event change the probability of the second? If not, the events are independent. If it does, they’re dependent, and the second probability must reflect what happened in the first.

With replacement

usually independent

Without replacement

usually dependent

Probability Review Quiz

Next: TSI Practice Test

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