10.1 Probability Basics

What is the probability of rolling a 4 on a fair six-sided die?

Solution:
Number of favorable outcomes = 1 (only one face shows 4)
Total number of outcomes = 6 (faces numbered 1 to 6)

$$P(\text{rolling a 4}) = \frac{1}{6}$$

As a decimal: $P = 0.167$ or about $16.7\%$

Answer: $\frac{1}{6}$ or approximately 0.167

A standard deck has 52 cards: 13 hearts, 13 diamonds, 13 clubs, 13 spades. What is the probability of drawing a heart?

Solution:
Number of hearts = 13
Total number of cards = 52

$$P(\text{heart}) = \frac{13}{52} = \frac{1}{4}$$

As a decimal: $P = 0.25$ or $25\%$

Answer: $\frac{1}{4}$ or 0.25

A factory makes light bulbs. Out of 500 bulbs tested, 15 were defective. Estimate the probability that a randomly selected bulb is defective.

Solution:
This is experimental probability (relative frequency):

Number of defective bulbs = 15
Total bulbs tested = 500

$$P(\text{defective}) = \frac{15}{500} = \frac{3}{100} = 0.03$$

Answer: 0.03 or 3%

Interpretation: About 3 out of every 100 bulbs are defective

If the probability of rain tomorrow is 0.3, what is the probability it won't rain?

Key Concept: $P(\text{not A}) = 1 - P(\text{A})$

Solution:
$P(\text{no rain}) = 1 - P(\text{rain})$
$P(\text{no rain}) = 1 - 0.3 = 0.7$

Answer: 0.7 or 70%

• Range: Probability is always between 0 and 1 (or 0% to 100%)
• All probabilities sum to 1: If you list all possible outcomes
• Fractions, decimals, percentages: All are valid ways to express probability
• Fair means equal chance: Fair die, fair coin, etc.
• Relative frequency gets better with more trials: Law of large numbers

List the sample space for flipping a coin once.

Sample Space: S = {H, T}

Number of outcomes: 2

If the coin is fair:
$P(H) = \frac{1}{2}$ and $P(T) = \frac{1}{2}$

Note: $P(H) + P(T) = \frac{1}{2} + \frac{1}{2} = 1$ ✓

List the sample space for rolling a six-sided die.

Sample Space: S = {1, 2, 3, 4, 5, 6}

Number of outcomes: 6

Example events:
• Even number: E = {2, 4, 6}, so $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$
• Greater than 4: E = {5, 6}, so $P(> 4) = \frac{2}{6} = \frac{1}{3}$

List all possible outcomes when flipping two coins.

Systematic listing:
First coin H: HH, HT
First coin T: TH, TT

Sample Space: S = {HH, HT, TH, TT}

Number of outcomes: 4

Find probabilities:
• Both heads: $P(HH) = \frac{1}{4}$
• Exactly one head: Events {HT, TH}, so $P = \frac{2}{4} = \frac{1}{2}$
• At least one head: Events {HH, HT, TH}, so $P = \frac{3}{4}$

How many outcomes are there when rolling two dice? Find P(sum = 7).

Sample Space:
For each outcome on first die (1-6), there are 6 outcomes on second die
Total outcomes = 6 × 6 = 36

Finding P(sum = 7):
List all ways to get sum of 7:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

That's 6 favorable outcomes

$$P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6}$$

Answer: $\frac{1}{6}$ or approximately 0.167

The letters of the word MATHS are written on cards. One card is chosen at random. Find the probability of choosing a vowel.

Sample Space: S = {M, A, T, H, S}

Number of outcomes: 5

Vowels: Only A is a vowel
Number of vowels = 1

$$P(\text{vowel}) = \frac{1}{5}$$

Answer: $\frac{1}{5}$ or 0.2

• Be systematic: List outcomes in an organized way to avoid missing any
• Two events: Use a table or tree diagram to list all combinations
• Count carefully: Double-check you haven't counted anything twice
• Product rule: If event A has m outcomes and event B has n outcomes, together they have m × n outcomes
• Check your work: All probabilities should add up to 1