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Data Handling - Probability Scales and Likelihood of Events

Grade 5IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Probability is a numerical measure of the likelihood that a specific event will occur, ranging from a value of 00 to 11. In a visual probability scale, this is represented by a straight horizontal line where 00 is at the far left and 11 is at the far right.

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The Likelihood Scale uses specific vocabulary to describe events: 'Impossible' occurs at 00, 'Unlikely' falls between 00 and 0.50.5, 'Even Chance' (or 'Equally Likely') is exactly at 0.50.5 (the midpoint), 'Likely' falls between 0.50.5 and 11, and 'Certain' occurs at 11.

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Probabilities can be expressed in three mathematical forms: as a fraction like 12\frac{1}{2}, as a decimal like 0.50.5, or as a percentage like 50%50\%. Visually, these all occupy the same point on the probability number line.

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Equally likely outcomes occur when every possible result in an experiment has the same chance of happening. For example, a fair six-sided die has six equally likely outcomes, each with a probability of 16\frac{1}{6}.

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A 'Fair' vs 'Unfair' experiment: A visual example of fairness is a spinner divided into identical, equal-sized pie slices. If the slices are different sizes (angles), the spinner is unfair because the outcomes do not have an equal chance of being selected.

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The sum of the probabilities of all possible outcomes in an experiment must always equal 11. For instance, if you look at a pie chart representing all outcomes, the total area of all slices combined represents the certain probability of 11.

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The 'Complement' of an event is the probability of that event NOT happening. If the probability of an event is PP, the probability of the complement is 1−P1 - P. Visually, if one part of a bar is shaded to show 'Success', the unshaded part represents 'Failure'.

📐Formulae

P(Event)=Number of favorable outcomesTotal number of possible outcomesP(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

P(Not Event)=1−P(Event)P(\text{Not Event}) = 1 - P(\text{Event})

Total Probability=∑P(all outcomes)=1\text{Total Probability} = \sum P(\text{all outcomes}) = 1

💡Examples

Problem 1:

A bag contains 33 red marbles, 22 blue marbles, and 55 green marbles. If you pick one marble without looking, what is the probability of picking a blue marble? Describe its likelihood on the probability scale.

Solution:

Step 1: Calculate the total number of outcomes by adding all marbles together: 3+2+5=103 + 2 + 5 = 10. \nStep 2: Identify the number of favorable outcomes (blue marbles), which is 22. \nStep 3: Apply the formula P(Blue)=210P(\text{Blue}) = \frac{2}{10}. \nStep 4: Simplify the fraction to 15\frac{1}{5} or convert to a decimal: 0.20.2. \nStep 5: Since 0.20.2 is between 00 and 0.50.5, the likelihood is 'Unlikely'.

Explanation:

We first find the total possibilities to create our denominator and then use the specific event count as our numerator. The result 0.20.2 is closer to impossible than to even chance, making it unlikely.

Problem 2:

You spin a fair spinner that is divided into 88 equal sections. 44 sections are yellow, 22 are red, and 22 are blue. What is the probability of the spinner landing on a color that is NOT red? Express your answer as a percentage.

Solution:

Step 1: The total number of equal sections is 88. \nStep 2: Find the number of sections that are NOT red. These are the yellow and blue sections: 4+2=64 + 2 = 6. \nStep 3: Write the probability as a fraction: P(Not Red)=68P(\text{Not Red}) = \frac{6}{8}. \nStep 4: Simplify the fraction: 6÷28÷2=34\frac{6 \div 2}{8 \div 2} = \frac{3}{4}. \nStep 5: Convert the fraction to a percentage: 34=0.75=75%\frac{3}{4} = 0.75 = 75\%.

Explanation:

This problem uses the concept of the complement or subtraction. We can either count the non-red sections directly or subtract the red sections from the total (8−2=68 - 2 = 6).