Probability

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Probability

• Vall Rasaratnam

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Basic rule of probability

• Probability is the study of random or
  non-deterministic experiments
• If you flip a coin there are two possible
  outcomes the coin could land on heads (H) or on
  tails(T)
• The set of outcomes is therefore H, T
• The set of all outcomes is called sample space

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Basic rules of probability

• Suppose we flip a coin and flip it again then the
  possible outcomes are HH, HT, TH, TT
• We might be interested in calculating the
  probability of getting HH (head followed by head)
  or (HT) or (TH)
• Each of these possible results that we are
  interested in is an event and each event is a
  subset of the sample space

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Events

• The probability of an event E is written as P(E)
• P(E) no of ways that an event can occur
• no of ways samples space can occur
• If an event is impossible its probability is 0
• If an event can occur for certainty then its
  probability is 1

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• So in the toss of a coin the probability of heads
  (H) is P(H) ½
• and probability of tails (T) is P(T) 1/2

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Events

• Suppose we toss a dice what is the probability
  of it landing on 1
• P(1) 1/6 as there is 6 possible outcomes
• What is the probability of getting an even number
• P(even number) 3/6 1/2

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Combined events

• Sometimes it is not easy to find the total number
  of possible outcomes of an event
• However we can represent all possible outcomes by
  using a tree diagram
• From tree diagram we can write down the number of
  desirable outcomes and total number of possible
  outcomes

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Example

• A coin is tossed 3 times. What is the
  probability of getting 2 heads?

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Tree diagram
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Independent events (AND)

• Events A and B are independent if one can happen
  whether or not the other has happened
• For example, on spinning a coin and throwing a
  dice, the events head and 6 are independent
• To find the probability that both A and B occur
  we use the product rule
• P(A and B) P(A) P(B)

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Example 1

• A card is drawn from a pack and a coin is spun.
  What is the probability of obtaining a spade and
  a head?

P(head) P(5) 13/52 ½ 13/104 1/8
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Example 2

• A dice and a coin are tossed. What is the
  probability of getting a head and a 5?

P(spade) P(head) ½ 1/6 1/13
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Example 3

• Two cards are drawn from a pack of 52 without
  replacement. What is the probability that they
  are an ace and a jack in that order

P(ace) P(jack) 4/52 4/51 16/2652 4/663
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Events

• Let a card be selected at random from a deck of
  52 cards. What is the probability that the card
  is
• Spade
• A face card
• Spade face card

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Two events occurring (OR)

• The probability that two events, A or B can occur
  is given by
• P(A or B) P(A) P(B) P(A and B)
• We use this when we want to find the probability
  of event A or probability of event B occurring
• By subtracting P(A and B), the probability of
  double counting is removed

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Example 1

• A card is drawn from a pack of 52. What is the
  probability that it is a king or a spade?
• P(king) 4/52
• P(spade) 13/52
• P(king and spade) 1/52
• P(king or spade) 4/52 13/52 1/52
• 4/13

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Example 2

• A card is drawn from a pack of 52. What is the
  probability of it being either a heart or a
  diamond?
• P(heart) 13/52
• P(diamond) 13/52
• P(diamond and heart) 0
• P(heart or diamond) 13/52 13/52 0
• 26/52 1/2

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Event not happening /At least once

• The probability that an event occurs or it does
  not occur is equal to 1
• P(E) P(not E) 1
• If x is the probability of an event happening,
  then the probability of it not happening is 1-x
• P(E) 1-P(notE) or
• P(Event happens at least once)
• 1 p(Event does not happen at all)

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Example 1

• For example when a dice is thrown P(6) is 1/6 so
  probability of not 6 is 1-1/6 5/6

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Example 2

• A bag contains 18 identical marbles, 8 of which
  are white and the remainder black. 3 marbles are
  removed at random, 1 at a time without
  replacement. Find the probability that
• All are black
• At least one is white

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• P(all black) 10/18 9/17 8/16
• 5/34
• P(at least one is white) 1-5/34
• 29/34

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Example 3

• The probability that a woman will hit a target
  with a single shot at a rifle range is 3/5. Find
  the probability that she first hits the target
  with her 3rd shot
• P(hit) P(miss) 1 (Either she hits it or
  misses it)
• P(hit) 3/5
• P(miss) 1 3/5 2/5

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• P(hit on 3rd shot) P(miss on 1st)
• P(miss on 2nd) P(hit on 3rd)
• 2/5 2/53/512/125

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Order of Events

• A bag contains 5 red sweets and 4 green ones.
  Pete takes 3 sweets out in succession and eats
  them.
• What is the probability tht
• (i) All 3 are red
• (ii) First 1 is red and next 2 green in that
  order
• (iii) One is red and two are green

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Order of Events

• P(all red) 5/9 4/83/75/42
• P(red and then green in that order
• 5/9 4/83/7 5/42
• P(one red and 2 green)
• 5/9 4/8 3/7 4/9 3/8 5/7 4/9 5/8
  3/7 180/504 5/14

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Order of event

• In the last example (iii) we did not know what
  order the sweets were drawn from the bag,
  therefore we can conclude that it could have
  happened in one of three ways
• 1st red, 2nd green,3rd green or
• 1st green, 2nd red, 3rd green