Or and And Problems

Discussion

• Or problems
  • In a simple experiment, this is a venn diagram
  • A six sided dice is tossed, What is the probability of

    	    a 2 or a odd number?
    	       P(E) =   P(2) + P(odd)  - P(2 and odd)
    	            = 1/6 + 3/6 - 0/6 
    		    = 4/6 = 2/3
    	        
    
    	    a number bigger than 3 or an odd number
    	       P(E) = P(n>3) + P(odd) - P(n>3 and odd)
    	            =  2/6 + 3/6 - 1/6 
    		    =  4/6 or 2/3
    	        
    
    	

  • If two events can not occur at the same time, the events are mutually exclusive, as in the first example.
• And problems
  • If a 6 sided dice is tossed and a coin is flipped, what is the probability of a 1 and a heads?

    	   There are 6*2 or 12 possibilities.
    	   P(1) = 1/6
    	   P(heads) = 1/2
               P(1 and H) = P(1) ×P(H) 
    	              = 1/6 × 1/2
    		      = 1/12
    	

  • What is the probability of an odd number and a tail?

    	P(odd and tail) = P(odd) × P(T):
    	                = 1/2 × 1/2
    			= 1/4
    	

  • The above events are independent events because the outcome of the first does not affect the outcome of the second.
  • Dependent events occur when the outcome of the first event effects the outcome of the second.
  • Sampling without replacement leads to dependent events.
• What is the probability of correctly selecting the winning power ball ticket?

         Select 5 numbers out of 55 without replacement
         Select 1 number out of 42
  
         Win by matching all numbers.
  
         The first set is an and problem.  These are dependent events.
         The second is independent of the first.
  
         P(1 right) = 5/55
         P(2 right) = P(1 right) * P(second right) 
                    = 5/55       *  4/54
         P(3 right) = 5/55 * 4/54 * 3/53
         P(all 5 right) = 5/55*4/54*3/53*2/52*1/51
         P(powerball) = 1/42
         P(win) = 5*4*3*2*1/(55*54*53*52*51) * 1/42 = 1/146,107,962
         

Homework