1 Probability

Probability is a way to find likeliness of an event to happen. Permutation and combination are ways to count events and possibilities. Statistics is used to summarize data.

It tells us:

  • how likely event is going to happen.
  • possibility of event that is fundamentally random.
  • Quantifying the uncertainity.

Any experiment can have outcome for an event. The possibility of outcome is the probability.

Experiment -> many events -> outcomes

Events make sample space, that is, all possible outcomes.

For example, if we flip a coin we can get either heads or tails. The possibility of heads is 50% and possibility ability of tails is 50%. So the probability of Heads is .5 and probability of tails is also .5, if it is a biased coin.

The analysis of event governed by probability is called statistics.

\[ P(e) = \frac{ Possibilities }{ Outcomes } \]

Theoretical or Classical Probability

  • It can be stated and seems fixed.
  • For example flipping a coin.

Experimental or Subjective Probability

  • Finding an outcome based on past data and experience
  • example prediction of the score. Probability gives a reasonable predictions about an outcome. It is highly likely but not hundred percent true.

Simulation and Randomness

  • We can use list of random numbers to simulate our experiment multiple times and average out to find confidence.

1.1 Events

Every possible outcome of a variable is an event.

Simple event

  • described by a single characteristic.
  • For eg, a day in January from all days in 2018.
  • Complement of an event A (denoted A’).
    • All events that are not part of event A.
    • For eg, all days from 2018 that are not in January.

Joint event

  • described by two or more characteristics.
  • For eg, a day in January that is also a Wednesday from all days in 2018.

Mutually Exclusive or Disjoint Sets

  • cannot occur simultaneously.
  • They have no intersection outcomes.
  • For eg, A = day in Jan, B = day in Feb. A and B cannot occur simultaneously.
    • In this, P(A1 U A2 U A3…) = P(A1) + P(A2) + P(A3)…
    • Also, P(A & B) = 0.

Collectively Exhaustive Events

  • One of the event must occur
  • The set of events covers the entire sample space
  • For eg, A = Weekday; B = Weekend; C = January; D = Spring;
    • Events A, B, C and D are collectively exhaustive (but not mutually exclusive – a weekday can be in January or in Spring).
    • Events A and B are collectively exhaustive and also mutually exclusive.

Independent Events

  • not dependent on each other. That is, occurrence of one does not affect occurrence of another event.

Note: All mutually exclusive events are dependent but not all dependent events are mutually exclusive.

1.2 Addition Rule

Addition rule of probability.

\[ P(A \cup B ) = P(A) + P(B) -P(A \cap B) \]

if mutually exclusive, then \(P(A \cap B) = 0\).

And is intersection, or is union.

For eg, P(Jan or Wed) = P(Jan) + P(Wed) - P(Jan and Wed) = 31/365 + 52/365 - 5/365 = 78/365

1.3 Multiplication Rule

For independent event, what happened in past event will have no effect on current event. For eg, P(HH) or P(at least 1H in 10 flips).

\[ P(HH) = 0.5 \times 0.5 \]

P(at least 1H in 10 flips) = 1 - P(All T in 10 flips)

\[ 1 - (0.5)^{10} = 1023 \div 1024 = 99.9% \]

1.4 Marginal or Unconditional Probability:

  • Simple probability like P(A) = 0.2, P(B) = 0.4

1.5 Joint Probability

  • P(A & B) both events to happen simultanieously

1.6 Conditional Probability

When we have to find a probability under a given condition.

Dependent Events

  • A|B is ‘A happening after B’ or ‘conditional prob of A given that B has occurred’.
  • B becomes the new sample space, because it’s A given B. Hence,

\[ P(A|B) = \frac{P(A \& B)}{P(B)} \]

Independent Events

  • if independent (does not affect each other), then

\[ P(A|B) = P(A) \]

Important outcome:

  • When finding P(A & B) we have to consider and analyse that whether A and B are dependent or not.
  • Based on dependency, our P(A & B) changes as follows:

If dependent, the probability of A and B is:

\[ P(A \& B) = P(A) \times P(B|A) = P(B) \times P(A|B) \]

else

\[ P(A \& B) = P(A) \times P(B) \]

because P(B|A) = P(B), occurrence of A has no effect on B.

Probability of A or B

\[ P(A \space or \space B) = P(A) + P(B) - P (A \& B) \]

Add all the joint probability of colectively exhaustive and mutually exclusive events to get marginal probability of one event.

eg, Consider industries and performance below.

| Poor |Avg|Good|Marginal —| small|0.02|0.07|0.01|0.1 medium|0.12|0.3|0.18|0.6 large|0.06|0.13|0.11|0.3 Marginal|0.2|0.5|0.3|1

1.7 Counting Events

1.7.1 Permutation

Arrange \(n\) people in \(k\) seats. To count number of ways in which this can be done we use permutation.

For eg, arrange 6 people in 3 seats, 6.5.4 = 6! / 3! = 120.

\[ _nP_k = \frac{n!}{(n - k)!} = n(n-1)... (k times) \]

Used when order matters and pick once (without replacement).

For eg,

\[ _{10}P_3 = 10.9.8 \]

1.7.2 Combinations

\[ _nC_k = \binom{n}{k} = \frac{_nP_k}{k!} = \frac{n!}{k!(n - k)!} = \frac{n(n-1)...[k \space times]}{k!} \]

We divide it by the number of ways in which k people can be arranged in k places, i.e, k! because ABCD and BCDA are same and we are counting this extra.

Order doesn’t matter, 123 = 312.

For eg,

\[ _{10}C_3 = \frac{10.9.8}{3.2.1} \]

1.8 Approach to solve a problem

We can take following approaches to solve a probability problem

  1. use simple definition, \[ P(e) = \frac{events \space possible}{sample \space space} \]

  2. Make a Contingency Table with possibilities.
    1. To find P(A or B), use P(A)+ P(B) - P(A and B)
    2. To find P(A and B), simply use (joint event)/total.
    3. To find P(A|B), P(A and B) / P(B)
  3. Make a Decision Tree, use when question has “after”.
    1. Find branches and outcomes
    2. Find effective value by multiplying with probabilities
    3. Roll back to find effective value at each branch.

  4. Use Venn Diagram when and/or is combined with not of a event.

  5. At least or at most, use \[ P(at least/most) = 1 - P(e) \]

Example

Find number of ways to arrange 1 - 10 digits in 3 places,

Repetition allowed, order matters = 10.10.10

Repetition not allowed, order matters = Permutation = 10.9.8

Repetition allowed, order doesn’t matter =

\[ \frac{10.10.10}{3.2.1} \]

Repetition not allowed, order doesn’t matter = Combination =

\[ \frac{10.9.8}{3.2.1} \]

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