11Aprobability

= Introduction to Probability =


 * __** Probability **__ is the analysis of the chances of something happening.


 * Pr(A) represents the probability that event A will occur.


 * Probability is measured on a scale of 0 to 1 where:
 * Pr(A) = 0
 * means A will never happen
 * 0 < Pr(A) < 0.5
 * means A is unlikely to happen
 * Pr(A) = 0.5
 * means A has an even chance of happening or not happening
 * 0.5 < Pr(A) < 1.0
 * means A is likely to happen
 * Pr(A) = 1.0
 * means A is certain to happen.


 * ** Examples: **
 * Pr(rain tomorrow) = 0.1
 * means it might rain tomorrow but the chances are low
 * Pr(a coin toss produces heads) = 0.5
 * means heads has an even chance of either happening or not happening
 * Pr(team A will win the game) = 0.75
 * means that team A is likely to win but it is certainly not guaranteed


 * Probability and Percentages **
 * Note that in the wider community, probability is often expressed as a percentage or as odds.
 * To change a probability to a percentage, multiply by 100
 * eg If Pr(A) = 0.36
 * Pr(A) = 0.36 × 100 = 36%
 * To change a percentage back to a probability divide by 100
 * eg If Pr(B) = 67%
 * Pr(B) = 67 ÷ 100 = 0.67


 * Combining Probabilities **


 * Pr(A È B) means Pr(either A __**or**__ B)
 * relate to A È B which means the Union or Combination of the A and B


 * Pr(A Ç B) means Pr(both A __**and**__ B)
 * relate to A Ç B which means the Intersection or overlap of A and B


 * ** Example **
 * A sport day is being organised among 50 students
 * 30 students wanted to play basketball (B)
 * 25 students wanted to play soccer (S)
 * these numbers include 10 students who wanted to play both
 * 5 students didn't want to play either
 * Find the probability that a randomly selected student
 * **(a)** . wants to play both basketball and soccer
 * **(b)** . wants to play either basketball or soccer
 * **(c)** . doesn't want to play basketball
 * **(d)** . wants to play soccer but not basketball


 * Solution
 * Creating a Venn Diagram such as that shown on the right is helpful when answering these types of questions.
 * **(a)** . wants to play both basketball and soccer
 * n(B Ç S) = 10
 * n( E ) = 50

math . \qquad \qquad \qquad Pr(B \, \cap \, S) = \dfrac{10}{50} = \dfrac{1}{5} math


 * **(b)** . wants to play either basketball or soccer
 * n(B È S) = 45
 * n( E ) = 50

math . \qquad \qquad \qquad Pr(B \, \cup \, S) = \dfrac{45}{50} = \dfrac{9}{10} math


 * **(c)** . doesn't want to play basketball
 * n(B ') = 50 – 30 = 20
 * n( E ) = 50

math . \qquad \qquad \qquad Pr(B\,') = \dfrac{20}{50} = \dfrac{2}{5} math


 * **(d)** . wants to play soccer but not basketball
 * n(B ' Ç S) = 15
 * n( E ) = 50

math . \qquad \qquad \qquad Pr(B\,' \, \cap \, S) = \dfrac{15}{50} = \dfrac{3}{10} math

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