Basic Probability Concepts

Why This Matters for Nursing: Probability helps you understand risk factors, test accuracy, and treatment outcomes. "There's a 20% chance of side effects" or "The test has a 95% accuracy rate" — understanding probability helps you communicate with patients and make informed decisions.

What You Need to Know

Probability measures how likely an event is to occur, expressed as a number between 0 and 1 (or 0% to 100%).

🧠 Memory Trick

Probability = Favorable ÷ Total

P(event) = (Number of ways it can happen) ÷ (Total number of possible outcomes)

"Part over Whole" — just like fractions and percents!

Example: Probability of rolling a 3 on a die = 1/6 (one 3 out of six sides)

Basic Probability Formula

P(Event) = Number of favorable outcomes / Total number of possible outcomes

Example: Drawing a Card

What's the probability of drawing a heart from a standard deck?

• Favorable: 13 hearts
• Total: 52 cards
• P(heart) = 13/52 = 1/4 = 0.25 = 25%

Probability Rules

Complement Rule

The probability something DOESN'T happen = 1 - P(it happens)

P(not A) = 1 - P(A)

Example: If P(rain) = 30%, then P(no rain) = 70%

Addition Rule (OR)

For mutually exclusive events (can't happen at same time): P(A or B) = P(A) + P(B)

Example: P(rolling 1 OR 2) = 1/6 + 1/6 = 2/6 = 1/3

Multiplication Rule (AND)

For independent events (one doesn't affect the other): P(A and B) = P(A) × P(B)

Example: P(heads AND heads) = 1/2 × 1/2 = 1/4

✏️ Worked Examples

Example 1: Simple Probability

Problem: A bag has 3 red, 5 blue, and 2 green marbles. What's the probability of drawing a blue marble?

Step 1 — Count the favorable outcomes. We want blue marbles. There are 5 blue marbles.

Step 2 — Count the total possible outcomes. Total marbles = 3 + 5 + 2 = 10.

Step 3 — Set up the fraction. P(blue) = favorable ÷ total = 5/10.

Step 4 — Simplify. 5/10 = 1/2 = 0.5 = 50%.

Answer: 50% chance of drawing a blue marble

Example 2: Complement Rule — Nursing Context

Problem: The probability a patient responds to treatment is 0.85. What's the probability they DON'T respond?

Step 1 — Use the complement rule. The complement rule says: if the probability of something happening is P, then the probability of it NOT happening is 1 - P. (Because something has to happen — all probabilities in a situation must add to 1.)

Step 2 — Calculate. P(no response) = 1 - 0.85 = 0.15.

Answer: 15% chance of no response

💡 Think about it: "it works" and "it doesn't work" are the only two options. If there's an 85% chance it works, there must be a 15% chance it doesn't — they add up to 100%.

Example 3: "Or" Probability

Problem: A die is rolled. What's the probability of rolling a 5 OR a 6?

Step 1 — Find each individual probability. A standard die has 6 faces (1 through 6).

• P(5) = 1/6
• P(6) = 1/6

Step 2 — Add them (for mutually exclusive events — things that can't happen at the same time). You can't roll both a 5 and a 6 on one roll, so they're mutually exclusive. P(5 or 6) = 1/6 + 1/6 = 2/6.

Step 3 — Simplify. 2/6 = 1/3 ≈ 0.333 ≈ 33.3%.

Answer: 1/3 (about 33.3%)

Example 4: "And" Probability — Independent Events

Problem: A coin is flipped twice. What's the probability of getting heads both times?

Step 1 — Identify that these are independent events. Independent means the first flip doesn't affect the second flip (the coin has no memory).

• P(heads on first flip) = 1/2
• P(heads on second flip) = 1/2

Step 2 — Multiply (for independent "AND" events). P(heads AND heads) = 1/2 × 1/2 = 1/4.

Answer: 1/4 (25%)

💡 Why multiply? Think of the possibilities: HH, HT, TH, TT. That's 4 equally likely outcomes, and only 1 of them (HH) is what we want. 1/4 = 25%. ✓

Example 5: Nursing Context

Problem: In a clinical trial, 45 out of 150 patients experienced side effects. What's the probability of side effects?

Step 1 — Identify favorable and total. Favorable (had side effects) = 45. Total patients = 150.

Step 2 — Calculate probability. P(side effects) = 45/150.

Step 3 — Simplify. Both divide by 15: 45÷15 = 3, 150÷15 = 10. So 3/10 = 0.30 = 30%.

Answer: 30% probability of side effects

Expressing Probability

*These all describe a one-in-four probability. True odds in favor are 1:3.

💡 Pro Tips

• Probability can never be negative or greater than 1 — if you get that, you made an error
• All probabilities in a situation must add to 1 (something has to happen!)
• "At least one" problems: Often easier to calculate 1 - P(none)
• Independent vs. dependent: Drawing WITH replacement = independent; WITHOUT replacement = dependent
• On the TEAS: Most questions are basic — counting favorable outcomes over total

⚠️ Common Mistakes to Avoid

• Probability > 100%: Impossible! Recheck your calculation
• Adding when you should multiply: "A AND B" = multiply; "A OR B" = add
• Forgetting to count total outcomes correctly: A die has 6 sides, a deck has 52 cards
• Confusing probability with odds: Probability 1/4 ≠ odds of 1:4 (odds are 1:3)

Quick Reference

Core Formula

P(Event) = Favorable outcomes ÷ Total outcomes

Key Rules

Probability Scale

0 ─────────── 0.5 ─────────── 1
Impossible    Even chance    Certain

Probability conquered! 💪 Next up: TEAS Math Practice Test