1A.) find the probability of the distribution for this experiment. In other words find P(T=x) for all possible values of x. 1B.) find the expected value of T, ie find E(T).

Probability Distributions 41 Now suppose that a student takes the test and arbitrarily guesses the an- swer to each question. A probability distribution table for this experiment is given below with an additional column to help us compute the expected value. 109 Sni dazoody X P(T = x) x . P(T = x) 0 (2/3)4 = 16/81 0 asks 1 4(1 /3) (2/3) = 32/81 32/81 2 6(1/3)2(2/3)2 = 24/81 48/81 3 4(1/3) (2/3) = 8/81 24/81 4 (1/3)4 = 1/81 4/81 Jail So the expected value is E(T) = 0+ 32/81 + 48/81 + 24/81 + 4/81 = 108/81 ~ 1.33. This means that if several people were to take this test by guessing the answers, we should expect the average number of correct answers for everyone to be about 1.33. Example 63. Expected values come up a lot in gambling. Here is a simple example. Suppose you choose one card from a deck of 52 playing cards. You make a deal with a friend that if the card drawn is a king, he will pay you $50. If the card drawn is not a king, you will pay him $10. We will compute your expected loss on this bet. Event X P(T = x) x . P(T = x) You win $50 4/52 $3.85 You lose -$10 48/52 $9.23 E(T) = -$5.38. Thus, if you were to play this game over and over, in the long run, you should expect to lose $5.38 per game. Not good for you. 6.2 Discrete Probability Distribution Exercises 1. Suppose you have a weighted coin that is twice as likely to land on heads as it is tails. You decide to flip this coin twice and record the number of times the coin lands on heads. Let T represent the outcome of this experiment. (a) Find the probability distribution for this experiment. In other words, find P(T = x) for all possible values of x. Daniel J. Geiger