Probability 2

Probability 2

1. The Multiplication Law.

   • If two events happen one after the other then the probability that A happens followed by B is given by P(A n B) = P(A) . P(B|A)
     • This means that the probability that A and then B happen is the product of the probability that A happens and the probability that B happens (given that A happened first)
       • For example if you picked a card out of a deck and didn't replace it and what is the probability that you get a King followed by a Queen
         • Probability (King then a Queen) = P(King) . P(Queen| First card was a King)
           • 4/52 X 4/51 = 4/663
   • This is an extenstion to the multiplication of counting.

2. Independent Events

   • This is when the occurrence of one event in no way affects the likelihood of occurrence of the other.
     • For Example if you flip a coin and then rolled a dice. The fact that you get a head on the coin has no baring on the number you will roll on the dice
   • A and B are independent if:
     • P(A) = P(A|B)
     • P(B) = P(B|A)
     • P(A n B) = P(A)P(B)

3. Expected Value

   • This is for when you want to decide is a bet is fair, bad or good.
     • Fair = 0
     • Good = >0
     • Bad = <0
   • For Example
     • E(X) = 2(1/2) + 4(1/4) + 6(1/4)
       • = €3.50
         • This means if you paid €5 its a bad bets as the result is €-1.50

4. Binomial Distribution : Bernoulli Trials

   • A binomial trial is a trail whose outcome is random and is one of just two possibilities which are called success and failure. Each trail is often referred to as a Bernoulli trial after the Swiss mathematician Jakob Bernoulli.
     • This is for when you want a number of sucess and a number of failures but its does not matter the order in which they resulted in.
   • • Must fit these criteria
       • There must be a finite number of trials
       • There are only 2 outcomes: Success and Failure
       • Trails are independent of each other
       • The Probability of Success (p) is the same for each trial.
     • For Example: There are 300 girls are 100 boys in a school, each day a teacher chooses one to make the morning announcements. What is the probability that in a given weeek she chooses 3 girls are 2 boys.
       • You might think that its
         • This would be wrong as it only accounts for GGGBB
           • While there is a range of solutions such as GGBGB or BBGGG or GBGBG
       • The correct solutions is
     • p = probability of sucess
       • n = no. of experiments
         • r = No. of wanted successes
   • Binomial Distribution Extended
     • This is for when you want to end within a certain period of time.
     • For Example you trow a dice until 6 is thrown 3 times. What is the probability that this takes 15 throws
       • Solution

5. Normal Distribution

   • Use your formula book to find out the probability from the normal.

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