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Data Handling - Introduction to Probability

Grade 6IB

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

🔑Concepts

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Probability is the numerical measure of the likelihood that a specific event will occur, expressed as a value ranging from 00 to 11. A probability of 00 means the event is impossible, while a probability of 11 means the event is certain.

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The Probability Scale is a visual tool represented as a straight horizontal line. On the far left is 00 (Impossible). In the exact center is 0.50.5 or 12\frac{1}{2}, representing an 'Even Chance' (like a fair coin toss). On the far right is 11 (Certain). You can describe the regions between these points as 'Unlikely' (between 00 and 0.50.5) and 'Likely' (between 0.50.5 and 11).

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An Experiment is any activity with an observable result, such as rolling a die or spinning a color wheel. The Sample Space is the set of all possible outcomes of that experiment. For a standard six-sided die, the sample space is visually listed as S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}.

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An Event is a subset of the sample space consisting of one or more outcomes. For example, if you are looking at a spinner divided into 8 equal parts, an event might be 'landing on a prime number', which would include the outcomes {2,3,5,7}\{2, 3, 5, 7\}.

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Theoretical Probability is calculated when all outcomes in a sample space are equally likely. It is a ratio that compares the number of successful outcomes to the total number of possible outcomes. It tells us what should happen in an ideal scenario.

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The Complement of an event consists of all the outcomes in the sample space that are NOT the event. Visually, if you imagine a Venn diagram circle representing event AA, the complement A′A' is everything outside that circle. The sum of the probability of an event and its complement is always 11.

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Relative Frequency, or Experimental Probability, is based on actual trials. Unlike theoretical probability, it uses data from a frequency table. For example, if you record the results of 100100 coin flips in a tally chart and get 5252 heads, the relative frequency of heads is 52100=0.52\frac{52}{100} = 0.52.

📐Formulae

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

P(A)+P(not A)=1P(A) + P(\text{not } A) = 1

P(not A)=1−P(A)P(\text{not } A) = 1 - P(A)

Relative Frequency=Frequency of eventTotal number of trials\text{Relative Frequency} = \frac{\text{Frequency of event}}{\text{Total number of trials}}

0≤P(A)≤10 \le P(A) \le 1

💡Examples

Problem 1:

A fair six-sided die is rolled. What is the probability of rolling a number greater than 44?

Solution:

  1. List the total outcomes (Sample Space): S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}. So, n(S)=6n(S) = 6.
  2. Identify the favorable outcomes (numbers >4> 4): {5,6}\{5, 6\}. So, n(E)=2n(E) = 2.
  3. Apply the probability formula: P(>4)=n(E)n(S)=26P(> 4) = \frac{n(E)}{n(S)} = \frac{2}{6}.
  4. Simplify the fraction: 26=13\frac{2}{6} = \frac{1}{3}.

Explanation:

To solve this, we first determine the total number of possible results on the die. We then identify which specific results meet our condition (being greater than 4). Finally, we express this as a simplified fraction.

Problem 2:

A bag contains 44 red, 55 blue, and 1111 white marbles. If one marble is picked at random, find the probability that it is NOT red.

Solution:

  1. Calculate the total number of marbles: 4+5+11=204 + 5 + 11 = 20.
  2. Find the number of red marbles: 44.
  3. Calculate the probability of picking a red marble: P(red)=420=15P(\text{red}) = \frac{4}{20} = \frac{1}{5}.
  4. Use the complement formula to find P(not red)P(\text{not red}): P(not red)=1−P(red)=1−15=45P(\text{not red}) = 1 - P(\text{red}) = 1 - \frac{1}{5} = \frac{4}{5}. Alternatively, add the non-red marbles: 5 (blue)+11 (white)=165 \text{ (blue)} + 11 \text{ (white)} = 16. P(not red)=1620=45P(\text{not red}) = \frac{16}{20} = \frac{4}{5}.

Explanation:

This problem uses the concept of 'complementary events'. You can either subtract the probability of the unwanted event from the total (1) or sum the probabilities of all other allowed events.