Why this matters for FAST: Probability problems appear frequently on FAST, requiring students to express probabilities as fractions, decimals, and percents, use sample spaces, and interpret what probabilities mean in context.
Students sometimes calculate probabilities greater than 1 or 100%, not realizing probability must be between 0 and 1.
"Probability is always between 0 (impossible) and 1 (certain). If you get a number greater than 1, you made a calculation error. Check that your favorable outcomes don't exceed total outcomes!"
Students think experimental results should always match theoretical predictions exactly.
"Theoretical probability is what SHOULD happen based on math. Experimental probability is what ACTUALLY happens when you try it. They get closer as you do more trials, but they won't always match exactly!"
Students think past results affect future probabilities (e.g., "I've flipped heads 5 times, so tails is due!").
"Each flip is INDEPENDENT - the coin doesn't remember what happened before! The probability of heads is always 1/2, no matter what happened on previous flips. Past results don't change future probabilities."
Show a coin. Ask: "If I flip this coin, what could happen?" (Heads or tails) "What are the chances of getting heads?" Lead to the idea that probability measures how likely something is to happen.
"The sample space is the set of ALL possible outcomes. For a coin flip, the sample space is {Heads, Tails}. For rolling a die, it's {1, 2, 3, 4, 5, 6}. We need to know all possibilities before we can find probability!"
Sample Space Examples
Coin flip: {H, T} - 2 outcomes
Die roll: {1, 2, 3, 4, 5, 6} - 6 outcomes
Spinner (4 colors): {Red, Blue, Green, Yellow} - 4 outcomes
Probability Formula
P(event) = Number of favorable outcomes / Total number of outcomes
Example: P(rolling a 3) = 1/6
Example: P(rolling even) = 3/6 = 1/2
"Count the outcomes you WANT (favorable) and divide by ALL possible outcomes (total). Probability can be written as a fraction, decimal, or percent!"
Probability Scale
0 = Impossible | 0.25 = Unlikely | 0.5 = Equally likely | 0.75 = Likely | 1 = Certain
P = 0: Rolling a 7 on a standard die
P = 0.5: Getting heads on a coin flip
P = 1: Rolling a number less than 7 on a die
Work through these together:
"A spinner has 8 equal sections: 3 red, 2 blue, 2 green, and 1 yellow. What is the probability of landing on blue or green?"
Solution:
P(blue or green) = (2 + 2) / 8 = 4/8 = 1/2 = 0.5 = 50%
For struggling students: Use hands-on manipulatives (coins, dice, spinners). Focus on simple one-event probabilities. Provide a probability number line visual.
For advanced students: Introduce compound events with tree diagrams. Challenge with problems involving "and" vs "or" probability. Have them design their own probability experiments.
For home: Send Parent Activity sheet. Families can explore probability with coins, dice, and cards.