Probability

What is probability, and what is its valid range of values?

Probability measures the likelihood of an event occurring. Value Range: Always between 0 and 1 inclusive (written as a fraction, decimal, or percentage from 0% to 100%). Rule: A probability can NEVER be negative or greater than 1.

What are the key benchmark values on the probability scale?

0 (0%) = Impossible | 0.25 (25%) = Unlikely | 0.5 (50%) = Even Chance | 0.75 (75%) = Likely | 1 (100%) = Certain

What is a sample space, and what is the formula for theoretical probability?

A sample space is a complete list, table, or grid of all possible outcomes. Formula: P(Event) = (Number of successful outcomes) / (Total number of possible outcomes).

How do you find P(Sum of 7) when rolling two 6-sided dice using a sample space?

Grid showing all 36 outcomes when rolling two 6-sided dice. Outcomes with a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) -> 6 outcomes total. P(Sum of 7) = 6/36 = 1/6.

What is a two-way table, and what is it used for in probability?

A grid used to categorize discrete data based on two variables (e.g., gender vs. transport method) to easily calculate overall or conditional probabilities.

In a two-way table example, how are data categories for boys and girls typically structured?

Boys: 18 Bus, 12 Walk (Total: 30) | Girls: 15 Bus, 25 Walk (Total: 40) | Totals: 33 Bus, 37 Walk (Grand Total: 70).

How do you calculate overall vs. conditional probability from a two-way table?

Overall P(Girl who walks) = 25/70 = 5/14. Conditional P(Walks given Girl) = (Girls who walk) / (Total girls) = 25/40 = 5/8.

What does it mean for two events to be mutually exclusive?

Events that CANNOT happen at the same time (e.g., rolling a single die and landing on both 2 and 5).

What is the addition rule for mutually exclusive events?

P(A or B) = P(A) + P(B). Example: If P(Red) = 0.3 and P(Blue) = 0.4, then P(Red or Blue) = 0.3 + 0.4 = 0.7.

What are exhaustive outcomes, and how do you use the complementary rule P(not A)?

Exhaustive outcomes cover ALL possible results; their probabilities sum to 1. Rule: P(not A) = 1 - P(A). Example: If P(Red) = 0.15 and P(Blue) = 0.45, then P(Yellow) = 1 - (0.15 + 0.45) = 0.40.

What is experimental probability (relative frequency), and what is its formula?

Probability based on actual trial data rather than theoretical prediction. Formula: Relative Frequency = (Number of successful trials) / (Total number of trials).

How does sample size affect the reliability of experimental probability?

Larger sample sizes (more trials) yield more reliable results because the experimental probability gets closer to the theoretical probability.

What is expectation (expected frequency), and how do you calculate it?

The predicted number of times an outcome will happen in future trials. Formula: Expected Frequency = Probability x Total number of trials. Example: If P(Heads) = 0.65, expected heads in 300 flips = 0.65 x 300 = 195.