Probability

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عدد المتطلبات السابقة الإلزامية (حتى الأساس): 6

ستتعلّم

• List sample spaces for simple experiments, using tables or tree diagrams for two stages.
• Compute probabilities of events with equally likely outcomes as fractions.
• Compare theoretical probabilities with relative frequencies from repeated trials or simulations.

تعريف

• A sample space is the set of possible outcomes of an experiment; an event is a subset of the sample space.
• If all outcomes in a finite sample space S are equally likely, the probability of an event A is P(A) = |A|/|S|.
• The relative frequency of an event in n trials is the number of times it occurred divided by n.

مبرهنة

• 0 ≤ P(A) ≤ 1, P(S) = 1, and P(not A) = 1 − P(A).
• As the number of independent trials grows, relative frequencies tend to settle near the probability (the law of large numbers, stated informally).

مفاهيم خاطئة شائعة

• The gambler's fallacy: thinking a fair coin is 'due' to land heads after several tails.
• Assuming all outcomes are equally likely, for example all sums of two dice.
• Believing a probability of 1/6 means the event happens exactly once in every 6 trials.

يقود إلى

اكتشافات اختيارية

• Measure-theoretic probability Kolmogorov (1933) placed probability on the foundation of measure theory.
• Markov chains A walker tosses a fair coin every minute: heads, one step east; tails, one step west. Is she certain to come back to where she started? What if she can also step north and south — or up and down in space?مسار الدهشة →
• ← Ratios, rates and proportional reasoning
• ← Counting principles

مسارات التعلّم

• Conventional K–12 progression
• Adult beginner pathway