Printable worksheet titled "Probability and Odds Worksheet" featuring exercises on determining the probability of events, using number lines, and calculating odds.

Here are the solutions to the problems on the worksheet, broken down step-by-step.

Part 1: Probability Scale

We need to match the events to the correct spot on the number line from 0 (Impossible) to 1 (Certain).

1. It will rain today: Weather is unpredictable. It might happen, it might not. This is in the middle. -> Likely (or Equally Likely depending on the forecast, but usually "Likely" or "Unlikely" fits best here. Given the options, let's look at the others first). *Correction:* Usually, without a forecast, this is treated as a 50/50 chance or just uncertain. Let's look at #2.
2. Today is Saturday: The problem states the current day is Tuesday (in the prompt instructions) or generally, a specific day cannot be another specific day. If today is Tuesday, it is Impossible for it to be Saturday. -> Impossible
3. You are in Algebra class: Assuming you are a student taking this test, this is happening right now. -> Certain
4. A coin will land tails up: A coin has two sides. Heads and Tails have an equal chance. -> Equally Likely
5. You will pass the next test: This depends on studying, but generally, students hope for this. However, compared to "Sun rising in the East," it's not certain. Compared to "Rain," it's similar. Let's re-evaluate based on standard textbook logic.
* *Standard Logic Check:*
* Impossible (0): Today is Saturday (if today is not Sat).
* Unlikely (< 0.5): You will go skiing tomorrow (unless you live in a ski resort, this is rare for most people).
* Equally Likely (0.5): Coin toss.
* Likely (> 0.5): It will rain today (often used as 'likely' if clouds are present, but ambiguous). Let's look at #6.
* Certain (1): You are in Algebra class (since you are doing the worksheet).
* *Let's refine:*
* 2. Today is Saturday -> Impossible (Assuming today is not Saturday).
* 4. Coin tails -> Equally Likely.
* 3. You are in Algebra class -> Certain.
* 6. You will go skiing tomorrow -> Unlikely (Most people don't ski daily).
* 1. It will rain today -> Likely (Often paired with unlikely in these sets).
* 5. You will pass the next test -> This is tricky. Usually, "Likely" is reserved for things with >50% chance. If we assume the student studies, it's likely. If we swap 1 and 5? "It will rain" is often 50/50. "Passing a test" is often considered "Likely" for good students. Let's stick with:
* 1: Likely
* 5: Likely (Wait, there is only one "Likely" slot? No, the scale has labels. Usually, these questions map to specific points. Let's assume standard mapping:
* Impossible: 2
* Unlikely: 6
* Equally Likely: 4
* Likely: 1 or 5
* Certain: 3
* *Self-Correction:* Actually, looking at the layout, there are 6 questions and 5 labels. Two questions might share a label, or one label is unused.
* Let's look at #5 again. "You will pass the next test." vs #1 "It will rain today."
* Let's look at #6 "You will go skiing tomorrow." This is definitely Unlikely for most.
* Let's look at #2 "Today is Saturday." If the worksheet is done on a Tuesday, this is Impossible.
* Let's look at #3 "You are in Algebra class." Certain.
* Let's look at #4 "Coin tails." Equally Likely.
* That leaves 1 and 5 for "Likely". Both fit "Likely" better than "Unlikely". I will assign both to Likely, or perhaps #1 is Unlikely if it's a sunny day? Without context, Likely is the safest bet for positive outcomes like passing tests or common weather events in these problems.

*Final Mapping for Part 1:*
1. Likely
2. Impossible
3. Certain
4. Equally Likely
5. Likely
6. Unlikely

Part 2: Find each probability if a die is rolled.

A standard die has 6 sides: {1, 2, 3, 4, 5, 6}. Total outcomes = 6.

7. P(3)
* There is only one "3" on the die.
* Probability = 1 out of 6.
* Answer: 1/6

8. P(even number)
* Even numbers on a die: 2, 4, 6.
* Count = 3.
* Probability = 3 out of 6.
* Simplify: 3/6 = 1/2

9. P(number less than 1)
* Numbers on a die start at 1. There are no numbers less than 1.
* Count = 0.
* Probability = 0 out of 6.
* Answer: 0

10. P(a number divisible by 4)
* Check numbers 1-6:
* 4 is divisible by 4.
* No other number (1,2,3,5,6) is divisible by 4.
* Count = 1 (just the number 4).
* Probability = 1 out of 6.
* Answer: 1/6

11. P(a number greater than 1)
* Numbers greater than 1: 2, 3, 4, 5, 6.
* Count = 5.
* Probability = 5 out of 6.
* Answer: 5/6

Part 3: Odds

*Note: Odds are written as "Successes : Failures" (or Favorable : Unfavorable).*

12. A number greater than 3
* Successes (Greater than 3): 4, 5, 6 (Count = 3).
* Failures (Not greater than 3): 1, 2, 3 (Count = 3).
* Odds = 3 : 3.
* Simplify: 1 : 1

13. A multiple of 2
* Successes (Multiples of 2): 2, 4, 6 (Count = 3).
* Failures (Not multiples of 2): 1, 3, 5 (Count = 3).
* Odds = 3 : 3.
* Simplify: 1 : 1

14. A number divisible by 3
* Successes (Divisible by 3): 3, 6 (Count = 2).
* Failures (Not divisible by 3): 1, 2, 4, 5 (Count = 4).
* Odds = 2 : 4.
* Simplify: 1 : 2

Part 4: Probability to Odds Conversion

15. If the probability of an event occurring is 2/3, what are the odds of the event occurring?
* Probability = 2/3 means:
* Favorable outcomes (Numerator) = 2
* Total outcomes (Denominator) = 3
* To find unfavorable outcomes (failures), subtract favorable from total: $3 - 2 = 1$.
* Odds = Favorable : Unfavorable
* Odds = 2 : 1
* Answer: 2 : 1

16. If the probability of an event occurring is 1/2, what are the odds that it will NOT occur?
* First, find the probability that it will NOT occur.
* $P(\text{not}) = 1 - P(\text{occurring}) = 1 - 1/2 = 1/2$.
* So, the probability of it NOT occurring is 1/2.
* Favorable outcomes for "NOT occurring" = 1.
* Total outcomes = 2.
* Unfavorable outcomes (meaning it DID occur) = $2 - 1 = 1$.
* Odds of it NOT occurring = Favorable (for not occurring) : Unfavorable (for not occurring).
* Odds = 1 : 1.
* Answer: 1 : 1

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Final Answer:

Part 1: Probability Scale
1. Likely
2. Impossible
3. Certain
4. Equally Likely
5. Likely
6. Unlikely

Part 2: Die Probabilities
7. 1/6
8. 1/2
9. 0
10. 1/6
11. 5/6

Part 3: Odds of Die Rolls
12. 1 : 1
13. 1 : 1
14. 1 : 2

Part 4: Converting Probability to Odds
15. 2 : 1
16. 1 : 1