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Statistics 9-12 / Week 10 / Monday
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Week 10 · Probability Rules

Monday

The lane draw as a sample space
// The lane draw, with and, or and not
⏱ about 20 min

Monday: The Lane Draw as a Sample Space

Eight numbered tokens rattle in the cloth bag. "Eight lane spots for the first heat," Comet says. "Fair draw."

"Before you pull one," Wren says, "what are all the things that could happen?"

"A 1, a 2, up to an 8," Comet says. "Eight outcomes, each as likely as the next."

Wren rules a grid on his clipboard, one box per token. "Now circle the even lanes."

Comet circles 4 boxes. "And the low lanes, 1 to 4." She draws a second loop around 4 boxes.

"What do you notice?" Wren asks. "Two boxes sit inside both loops."

Nova hovers with the Run Log glowing. "Would you like a hint? The whole grid is your sample space."

"And every question about the draw is a set of boxes," Comet says. "Or, and, not. Let us name them."

The lane draw: Comet lifts a numbered token from a cloth bag while Wren sketches a grid of outcomes.

The crew's lane bag

Every number this week is the crew's own reading from Nova's Run Log. The bag holds 8 tokens numbered 1 to 8, one lane spot each.

The sample space is the set of all outcomes: {1, 2, 3, 4, 5, 6, 7, 8}. Each token is equally likely, so each has probability 1/8.

Events as sets of outcomes

An event is a set of outcomes. "Even lane" is {2, 4, 6, 8}. "Low lane" (1 to 4) is {1, 2, 3, 4}.

"Even and low" keeps only the outcomes in both sets: {2, 4}. That is the intersection.

"Even or low" gathers every outcome in either set, counting shared ones once: {1, 2, 3, 4, 6, 8}. That is the union.

"Not even" is every outcome outside the even set: {1, 3, 5, 7}. That is the complement.

EventOutcomesCountProbability
even{2, 4, 6, 8}41/2
low (1 to 4){1, 2, 3, 4}41/2
even and low{2, 4}21/4
even or low{1, 2, 3, 4, 6, 8}63/4
not even{1, 3, 5, 7}41/2

A solved problem to study

Here is how Wren finds the probability that the drawn lane is even or low.

  1. List the even outcomes: {2, 4, 6, 8}, 4 of them.
  2. List the low outcomes: {1, 2, 3, 4}, 4 of them.
  3. Join the two lists, writing each outcome once: {1, 2, 3, 4, 6, 8}. The shared outcomes 2 and 4 appear once, not twice.
  4. Count the union: 6 outcomes.
  5. Divide by the size of the sample space: 6 ÷ 8 = 3/4.

A probability is a count over a count: the outcomes in the event over all the outcomes. Write it in lowest terms.

PROBABILITIES FROM THE LANE BAG
  • Read the question.
  • Tap your answer.
Tokens 1 to 8 are equally likely. Even = {2, 4, 6, 8} and low = {1, 2, 3, 4}. What is P(even and low)?
Tokens 1 to 8 are equally likely. Even = {2, 4, 6, 8} and low = {1, 2, 3, 4}. What is P(even or low)?
Tokens 1 to 8 are equally likely. Even = {2, 4, 6, 8}. What is P(not even)?
Tokens 1 to 8 are equally likely. Odd = {1, 3, 5, 7} and high = {6, 7, 8}. What is P(odd and high)?
NAME THE SET
  • Read the question.
  • Tap your answer.
"Even and low" is which kind of set?
"Not even" is which kind of set?
"Even or high" is which kind of set?
How many tokens are both even and low? Type the number.
WHY THIS EXERCISEThe intersection is the overlap of two sets. Every "and" question starts by counting it.
StatementTrue or false?
Every token belongs to exactly one outcome of the sample space.?
4 + 4 - 2 = 6?
P(not even) = 1 - P(even).?
The union of two events always has as many outcomes as the two counts added.?
WHY THIS EXERCISENaming sets exactly is what makes the fractions trustworthy. The arithmetic comes after.
Try it
Write 1 to 8 on index cards. Turn the even cards face up and slide the low cards to the left.
The face-up cards on the left are the intersection. Every card that is face up or on the left is the union.
Draw the 8 tokens in a row. Loop the even ones in one color and the low ones in another.

A clear start. Tomorrow the addition rule turns the "or" count into a formula, and you check it two ways.