Probability

Maths · Class 12

Simulation · Maths · Class 12

Shrink the sample space

"Shrink the sample space" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Conditional probability" in Probability. Each square is one equally likely outcome (first die, second die). Pick an event E and a condition F. Outcomes outside F turn grey: they can no longer happen. P(E|F) is the share of E among the squares that are left. A condition F throws away every outcome outside F and makes F the new sample space. For equally likely outcomes P(E|F) = n(E ∩ F)/n(F); in general it is P(E ∩ F)/P(F).

From the lesson Conditional probability in Probability. Change the values and watch what happens.

Shrink the sample spaceMaths · Class 12

The idea behind it

NCERT §13.1–13.2, Examples 1–3

  • When it is known that an event F has occurred, the outcomes outside F are no longer possible. F becomes the new sample space, and the probability of E is recomputed inside it. The result is the conditional probability of E given F, written P(E|F).
  • Three fair coins are tossed: 8 equally likely outcomes. E: at least two heads = {HHH, HHT, HTH, THH}; F: the first coin shows tail = {THH, THT, TTH, TTT}. P(E) = P(F) = 1/2 and E ∩ F = {THH}, so P(E ∩ F) = 1/8.
  • Given F, only its four outcomes remain, and just one of them (THH) lies in E. So P(E|F) = 1/4, not 1/2: the information has changed the probability.
  • With equally likely outcomes, P(E|F) = n(E ∩ F)/n(F). Dividing top and bottom by n(S) gives the form used in general.
  • Definition: P(E|F) = P(E ∩ F) ÷ P(F), which needs P(F) ≠ 0.
  • With P(A ∩ B) = 4/13 and P(B) = 9/13, the definition gives P(A|B) = (4/13) ÷ (9/13) = 4/9. The value P(A) = 7/13 given alongside is not needed for this.
  • A family has two children, so S = {bb, bg, gb, gg}. Given that at least one is a boy (3 outcomes), the probability that both are boys is 1/3.
  • Cards numbered 1 to 10; one is drawn and is known to be more than 3, leaving {4, 5, …, 10}. Four of these seven are even, so P(even | more than 3) = 4/7.
  • P(E|F) and P(F|E) are different quantities: they share a numerator but are divided by P(F) and P(E) respectively.

Questions

2 answers

What does the "Shrink the sample space" simulation show?
Each square is one equally likely outcome (first die, second die). Pick an event E and a condition F. Outcomes outside F turn grey: they can no longer happen. P(E|F) is the share of E among the squares that are left.
Is this simulation free?
Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Conditional probability" in Probability (Class 12 Maths).

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