Probability
A bag has 4 red and 6 blue marbles. What is P(red)?
P(red) = 4/10 = 2/5 = 40%
What is the slope of a line through (0, 2) and (3, 8)?
m = (8−2)/(3−0) = 6/3 = 2
A right triangle has legs 5 and 12. Find the hypotenuse.
5² + 12² = 25 + 144 = 169 → c = 13
In y = a · bˣ, what does 'b' represent?
b is the growth factor — what you multiply by at each step. b > 1 = growth, 0 < b < 1 = decay.
You have 4 shirts, 3 pants, 2 shoes. How many outfits?
4 × 3 × 2 = 24 outfits.
You flip a fair coin 10 times and get heads every time. What is P(heads) on flip 11?
Exactly 1/2 = 50%. Each flip is independent. The coin has no memory.
Salaries: $30k, $35k, $38k, $40k, $200k. Which better describes typical pay — mean or median? Why?
Median ($38k). The mean is pulled up by the $200k outlier. Median is not affected by extreme values.
What is the distance between (0, 0) and (6, 8)?
d = √(6² + 8²) = √(36 + 64) = √100 = 10
Which is faster? Exponential growth or linear growth?
Exponential Growth
How many ways can 5 people line up in a row?
5! = 5 × 4 × 3 × 2 × 1 = 120 ways.
In the Monty Hall problem, you always switch after the host reveals a goat. What is your probability of winning?
2/3. Your door stays at 1/3; the other door gets the remaining 2/3 because the host's reveal is not random.
A line has equation y = −5x + 12. What is the slope, and what does the negative slope mean?
Slope = −5. The line goes down — for every 1 unit right, the value decreases by 5. Example: battery draining 5% per hour.
A ladder is 17 feet long. Its base is 8 feet from the wall. How high up the wall does it reach?
8² + h² = 17² → 64 + h² = 289 → h² = 225 → h = 15 feet
A culture starts with 50 bacteria and doubles every hour. Write the equation and find how many after 4 hours.
y = 50 · 2ˣ. After 4 hours: 50 × 2⁴ = 50 × 16 = 800 bacteria.
A committee of 3 is chosen from 8 people. Is this a permutation or a combination? Why?
It is a combination. Order does not matter.
A basketball player hits 6 shots in a row. According to research, what does this tell us about their next shot — and what is this called?
Almost nothing — their next-shot percentage is nearly identical to their baseline. This cognitive illusion is called the Hot Hand Fallacy.
The y-axis of a graph starts at 95 and ends at 100. What misleading trick is this?
Truncated y-axis.
How does the distance formula connect to the Pythagorean theorem? Explain and give an example.
The distance formula IS the Pythagorean theorem — the horizontal difference is leg a, the vertical difference is leg b, and d is the hypotenuse c. Example: (1,1)→(4,5): legs = 3 and 4, d = 5.
A population starts at 1 bacterium and doubles every hour. How many after 8 hours? No calculator — build it up.
1→2→4→8→16→32→64→128→256. Answer: 256 bacteria. (y = 1 · 2⁸ = 256)
What is the difference between P(6,3) and C(6,3)? Calculate both.
P(6,3) = 6×5×4 = 120 — ordered arrangements. C(6,3) = 120÷6 = 20 — unordered groups. C = P÷r! because each group can be arranged r! ways.
With 23 people in a room, what is the approximate probability that two of them share a birthday?
About 50%.
A car travels at constant speed. At t=2 hours it has gone 90 miles. At t=5 it has gone 225 miles. Write the equation and find its speed.
m = (225−90)/(5−2) = 135/3 = 45 mph. b: 90 = 45(2) + b → b = 0. Equation: y = 45x. The car started from the origin.
A right triangle has legs of 20 and 21. Find the hypotenuse without a calculator.
20² + 21² = 400 + 441 = 841. What squared is 841? 29² = 841. Hypotenuse = 29. (20-21-29 is a Pythagorean triple.)
A medicine starts at 400mg. Each hour, 25% is eliminated. Write the equation and find the amount after 3 hours.
y = 400 · (0.75)ˣ. After 3 hours: 400 × 0.75³ = 400 × 0.422 ≈ 168.75 mg.
A lottery picks 6 numbers from 1–49. How many possible tickets exist? And what is the probability of winning with one ticket?
Show your work. Partial credit is possible.
C(49,6) = (49×48×47×46×45×44)÷720 = 13,983,816. P(win) = 1/13,983,816 ≈ 0.0000000715 — about 1 in 14 million.