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Deal or No Deal

Deal or No Deal. $.01 $500 $800 $1000 $75000 $500000. 1. 2. 3. 4. 5. You are offered $60 000 for your case (number 6) Deal or No deal ?. 6. Random Variables. A random variable is a function. 

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Deal or No Deal

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  1. Deal or No Deal $.01 $500 $800 $1000 $75000 $500000 1 2 3 4 5 You are offered $60 000 for your case (number 6) Deal or No deal ? 6

  2. Random Variables • A random variable is a function.  • The randomness comes from not knowing which element of the domain that it will be evaluated at.

  3. E.G. 1: • Cups: • Win $50 if you guess the right cup and win nothing if you lose. It costs $20 to play. The random variable X denotes the gross amount that you could win. X={0,$50} is the set of all possible outcomes. This is the range of the function.

  4. Assignment • These values get assigned when a cup is picked, the cup choice is the domain of the function, in this example the ball is in Cup 2 This meets the criteria for a function as for each input value (cup choice) there Is ONLY one output value (pay-out X)

  5. our random variable X represents how much money you could win after playing the game. The probability distribution P is the set of probabilities associated with elements of the set X.

  6. Without a table, we can express the individual probabilities as P(X=0)=2/3 and P(X=50)=1/3 this notation is often shortened to    p(0)=2/3 and P(50)=1/3

  7. Expected Value • E(x) the expected value (or expectation) of a random variable is the average value that it will take if the experiment is repeated many times. This is the law of large numbers as applied to random variables. It can be calculated in a manner similarly to a weighted mean.

  8. Expectation • The Expected Value of X:

  9. For Cups = 0(2/3)  + 50(1/3) = 16.67

  10. Interpretation • This means that if you play this game many times, you will win an average of 16.67 each time that you play. As it costs 20 to play, you will lose money long term. All casino games follow this simple principal. It guarantees that the casino will always make money in the long run. 

  11. Back to Deal or No Deal = x1P(X=x1)+ x2P(X=x2)+ x3P(X=x3)+…+ xnP(X=xn)

  12. Deal or No Deal 1 2 $.01 $500 $800 $1000 $75000 $500000 3 4 5 You are offered $60 000 for your case (number 6) Deal or No deal ? 6 X= the amount of money in our box X= .01 or 500 or 800 or 1000 or 75000 or 500000 The probability that X=.01 is P(X=.01)= 1/6 This is the same for all of the others.

  13. X= .01 or 500 or 800 or 1000 or 75000 or 500000 • The probability that X=.01 is P(X=.01)= 1/6 • E(X) = x1P(X=x1)+ x2P(X=x2)+ x3P(X=x3)+…+ xnP(X=xn) • E(X) = .01 (1/6) + 500 (1/6) +800(1/6) + 1000(1/6) +75000(1/6)+500000(1/6) • =96216.67 No Deal

  14. Uniform Distribution • Consider the R.V. the number on the face of a die after rolling it: • X={1,2,3,4,5,6} • The probability associated with each outcome is the same: We say that the probability is uniformly distributed amongst all possible Outcomes.

  15. Uniform Distribution When probability is uniformly distributed each probability is 1/n. In this case p(x) =1/6 for each x in the set X.

  16. Practice • Page 374 1 to 13

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