L-PR-1 · Probability Evidence Lab
The numbers match. Does the evidence?
What is the most important difference between the two displays?
Anonymous check
Which claim matches the evidence?
Spinner A is fair and has 3 blue sectors out of 5. Spinner B produced blue 30 times in 50 recorded spins.
Decide first. Be ready to point to the evidence that supports your choice.
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Both values come only from recorded trials
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Spinner A uses visible equal possibilities; Spinner B uses observed outcomes
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Spinner A must also have been spun 50 times
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Spinner B is guaranteed to land blue on exactly 60% of future spins
After discussion: B — A uses a stated fair structure; B uses what happened in a set of trials. An observed rate is evidence, not a guarantee.
Inquiry map
Four questions will organize the lab.
Each question becomes useful because the previous evidence creates a new problem.
Build the possible outcomes before calculating.
Equal possibilities, observed data, or judgment?
OR gathers regions; AND selects shared outcomes.
Can they happen together? Does one change the other?
Words and visible regions come before a formula.
Anonymous check
What proportion studied in groups?
A survey contains 120 students. The table gives 60 who studied in groups and 60 who studied alone.
| Passed | Failed | Total | |
|---|---|---|---|
| Studied in groups | 42 | 18 | 60 |
| Studied alone | 30 | 30 | 60 |
| Total | 72 | 48 | 120 |
Decide first. Be ready to point to the evidence that supports your choice.
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\(0.25\)
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\(0.40\)
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\(0.50\)
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\(0.60\)
After discussion: C — \(60/120=0.50\). The denominator is all 120 surveyed students.
Empirical probability · observed evidence
Small batches vary. The combined record stabilizes.
Five groups each spin the same spinner 10 times.
Discuss: What stays fixed across the batches, and what varies?
Empirical probability is an observed relative frequency.
Sample space → event
Build the event before naming it.
Roll one fair six-sided die. Drag or point each possible outcome into “even” or “outside even.”
Discuss: Which outcomes belong to “roll an even number”?
Event A contains \(3\) of \(6\) equally likely outcomes, so \(P(A)=3/6=0.50\).
Anonymous check
Which probability claim is best supported?
A bent coin produced heads 68 times in 100 test tosses. No physical model proves that heads and tails are equally likely.
Decide first. Be ready to point to the evidence that supports your choice.
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Use \(1/2\) because every object called a coin is fair
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Use 0.68 as an empirical estimate for heads
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Use 68 because heads occurred 68 times
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Use 0 because heads and tails cannot occur on the same toss
After discussion: B — \(68/100=0.68\) is the observed relative frequency. Fairness was not established.
C4 · Optional evidence comparison · optional
Some probabilities come from informed judgment.
A new event has no equal-likelihood model and little or no repeated data.
Experience, mechanisms, and comparable situations can inform a probability.
The estimate should be presented as revisable judgment, not measured fact.
Judgment can be useful when its evidence, assumptions, and uncertainty are explicit.
Anonymous check
Which claim could be a probability?
A team reports four probability claims about a single event.
Decide first. Be ready to point to the evidence that supports your choice.
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\(-0.20\)
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\(0.37\)
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\(1.08\)
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\(140\%\)
After discussion: B — a probability must lie from 0 to 1 inclusive. The other values are below 0 or above 1.
Probability rules · earned from the evidence
Three audit rules catch many errors.
Use these only after identifying the possible outcomes and selected regions.
- Bound Never below 0
A negative share of possible or observed outcomes is impossible.
- Whole All possibilities total 1
The complete sample space accounts for certainty.
- No shared outcome Add without subtracting
This shortcut works only when the selected events cannot happen together.
A rule is a reasoned constraint, not a symbol to memorize.
Complement · same event space
The complement is everything outside the event.
Keep the die event A = “roll an even number.” Now focus on every possible outcome outside A.
Discuss: Which outcomes are not in A?
Together, an event and its complement fill the whole: \(P(A^\prime)=1-P(A)\).
Anonymous check
What is \(P(\text{not on time})\)?
A transit app estimates \(P(\text{on time})=0.78\) for tomorrow’s trip.
Decide first. Be ready to point to the evidence that supports your choice.
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\(0.12\)
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\(0.22\)
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\(0.78\)
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\(1.78\)
After discussion: B — \(1-0.78=0.22\). “On time” and “not on time” fill the whole without overlap.
Two events · meaning before notation
OR gathers regions. AND selects the shared region.
For one die, A means “even” and B means “greater than 3.”
Discuss: Where should each die outcome go?
Everyday words select visible regions before symbols name them.
Addition rule · derive it from the regions
Subtract the overlap once.
Of 120 students, 54 are in art club and 42 are in sports club. Eighteen are in both.
Discuss: Why is 54 + 42 too large for “art OR sports”?
For OR: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\).
Anonymous check
What is \(P(\text{tablet OR laptop})\)?
In a group of 100, 46 use a tablet, 38 use a laptop, and 14 use both.
Decide first. Be ready to point to the evidence that supports your choice.
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\(0.56\)
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\(0.70\)
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\(0.84\)
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\(0.98\)
After discussion: B — \((46+38-14)/100=70/100=0.70\).
Mutually exclusive events
Zero overlap creates a shortcut.
For one trip to school, A means “walk” and B means “take the bus.” No student does both on the same trip.
Discuss: What feature of the event space makes direct addition safe?
Events with no shared possible outcome are mutually exclusive.
Relationship court · same toss versus two tosses
Two different questions judge event relationships.
Can both happen? Does knowing one change the chance?
Outcomes: \(\mathrm{H}\) and \(\mathrm{T}\).
Can “heads” and “tails” occur together on this toss?
Equally likely: \(\{\mathrm{HH},\mathrm{HT},\mathrm{TH},\mathrm{TT}\}\).
Does the first toss being heads change the chance of heads on the second toss?
Discuss: Which asks “together”? Which asks “chance changes”?
Mutually exclusive and independent answer different questions.
Anonymous check
Which description is correct?
A coin model treats \(\{\mathrm{HH},\mathrm{HT},\mathrm{TH},\mathrm{TT}\}\) as four equally likely outcomes. Event A is “heads on toss 1.” Event B is “heads on toss 2.”
Decide first. Be ready to point to the evidence that supports your choice.
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A and B cannot happen together
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A and B can happen together, and knowing A does not change the chance of B
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A occurring makes B impossible
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A and B mean the same event
After discussion: B — \(\mathrm{HH}\) is shared. Among \(\mathrm{HH}\) and \(\mathrm{HT}\), one of two outcomes has a second head—the same \(1/2\) as in the full model.
Faded example → transfer
Translate a table into the same four regions.
A survey of 200 students records sports participation and volunteering.
Discuss: Where does each interior table cell belong on the event-space map?
| Volunteer | Not volunteer | Total | |
|---|---|---|---|
| Sports | 40 | 50 | 90 |
| Not sports | 30 | 80 | 110 |
| Total | 70 | 130 | 200 |
Sports OR volunteer: \((50+40+30)/200=0.60\); neither: \(80/200=0.40\).
Independent construction · cards 1–12
Build the exit event space.
Choose one card numbered 1 through 12. A means “even.” B means “a multiple of 3.”
Discuss: Place every card into A only, both, B only, or neither.
Keep this event space visible for the next three decisions.
Anonymous check
What is \(P(A\text{ OR }B)\)?
One card is chosen from 1 through 12. A = even. B = multiple of 3.
Decide first. Be ready to point to the evidence that supports your choice.
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\(1/3\)
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\(1/2\)
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\(2/3\)
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\(5/6\)
After discussion: C — eight cards are in A only, both, or B only, so \(8/12=2/3\).
Anonymous check
What is \(P(\text{neither }A\text{ nor }B)\)?
Use the same cards and events. A = even. B = multiple of 3.
Decide first. Be ready to point to the evidence that supports your choice.
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\(1/6\)
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\(1/3\)
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\(2/3\)
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\(5/6\)
After discussion: B — cards 1, 5, 7, and 11 are outside A OR B, so \(4/12=1/3\).
Anonymous check
Are A and B mutually exclusive?
Use the same cards and events. A = even. B = multiple of 3.
Decide first. Be ready to point to the evidence that supports your choice.
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Yes, because each card is selected only once
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Yes, because both events have several outcomes
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No, because cards 6 and 12 satisfy both events
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No, because \(P(A\text{ OR }B)<1\)
After discussion: C — the shared outcomes 6 and 12 prove that the events can happen together, so they are not mutually exclusive.
Synthesis
The decision path begins with evidence.
Use this route whenever a probability problem combines events.
List or identify all possible outcomes and the appropriate total.
Equal possibilities, observed trials, or informed judgment.
OR gathers; AND selects the shared region; NOT selects outside.
Can both happen? Does one change the chance of the other?
A formula is the last compressed line of a visible argument.
Exit reflection · confidence follows evidence
What can you now explain?
Use your three exit decisions as evidence. Then rate how confidently you could explain each idea.
Confidence is a reflection—not proof of mastery or a grade. Use it beside your three exit responses.
I can identify whether a probability uses equal possibilities, observed data, or informed judgment.
I can use probability bounds and complements to audit a claim.
I can translate OR and AND into regions and count overlap once.
I can distinguish “can happen together” from “changes the chance.”