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3.04 exPECTED VALUE Objective: Students determine the expected values of random variables arising from statistical experiments like the rolling of a dice, flipping of a coin, spinning a spinner etc. Recall how we get the expected payoff when we roll a pair of dice and use this polynomial: Look at the polynomial for tossing a pair of dice: (x + x2+ x3 + x4 + x5 + x6 )2 = 1x2+ 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 5x8 + 4x9 + 3x10 + 2x11 + 1X12 If you roll a sum of 3, I gave you 10 dollars, otherwise you pay me 5 dollars . So, Expected payoff for the sum of 3: Expected payoff = Expected gain – expected loss Probability of a sum of 3 = 2/36 Expected gain = 10 x 2/36 = 10 x .05 = .50 cents Probability for not having a sum of 3 = (36-2)/36 = 34/36 Expected loss = 5 x 34/36 = 5 x .94 = 4.70 dollars Expected payoff = .50 cents – 4.70 dollars = -4.20 dollars What about for every sum you roll you win the following and you loss nothing: 1x2+ 2x3 + 3x4 + 4x5 Roll a sum of + Win($) 5x6 + 6x7 + 5x8 + 4x9 + 3x10 + 2x11 + 1X12 Probability Expected payoff (Win x probability) 2 3 1 2 1/36=.03 2/36=.05 .03 .10 4 5 6 7 8 9 10 11 12 3 4 5 6 7 8 9 10 11 3/36 =.08 4/36 = .11 5/36 = .14 6/36 = .16 5/36=.14 4/36=.11 3/36=.08 2/36=.05 1/36=.02 .24 .44 .70 .96 .98 .88 .72 .50 .22 Total = $5.77 Expected value winning = total OF THE PROBABILITIES X WIN SEATWORK: ROLL THE DICE 5 TIMES. ALL SUMS YOU WIN 5 DOLLARS. BUILD THE TABLE 3.04a Homework/classwork – expected value Polynomial for rolling dice 5 times: Expand ( x + x2 + x3 + X4 + X5 +X6)5 = 65 = 7776 possible outcomes Roll a sum of Win($) 2 5 3 5 4 5 6 7 8 9 10 11 12 ……… 30 Expected value = 5 5 5 5 5 5 Probability 1/7776= 5 5 5 Expected value= Total of expected payoff Expected payoff (Win x probability) OTHER EXAMPLES EXAMPLE 1. Toss five coins. What is the expected value for the number of heads? Polynomial : ( H + T)5 = H5 + 5H4T + 10H3T2 +10 H2T3 5HT4 + T5 Total possible outcomes : 25 = 32 Number of heads(H) Probability Product: Number of heads x probability 0 1/32 0 x 1/32 = 0/32 1 5/32 1 X 5/32 = 5/32 2 10/32 2 x 10/32 = 20/32 3 10/32 3 x 10/32 = 30/32 4 5/32 4X 5/32 = 20/32 5 1/32 5 X 1/32 = 5/32 Total 80/32 Expected value = 0/32 + 5/32 + 20/32 + 30/32 + 20/32 + 5/32 = 0+5+20+30+20+5 32 = 80/32 or 2.5 Interpretation : On the average, you would expect that you would only get 2.5 heads in flipping 5 coins most of the time Example 3. The following gives the distribution of lotto betters who hit the winning numbers in the Mega Millions draw of Dec. 4, 2015. What is the expected number of winning numbers that was correctly hit during the draw? Numbers Correctly Frequency Product: 6 0 6x0= 0 5 0 5x 0= 0 4 4 4x4 = 16 3 233 3 x233 = 699 2 7756 Matched 2 x 7756 = 15,512 1 161,506 1 x 161,506 = 161506 0 1,234,000 0 Total 1, 403,499 178033 178,033 Expected Value = 1,403,499 = .12 Interpretation: On the average, we expect that only .12 or 0 Winning numbers will be hit. It’s always a win for the state lottery most of the time in this draw. Definition – RANDOM VARIABLE A RANDOM VARIABLE IS ANY OUTCOME THAT IS an INPUT TO get A NUMBER OR VALUE, LIKE PROBABILITY, number of ways, OR EXPECTED VALUE. It is usually denoted by a letter like x, y, h, t, etc. Input Outcomes Rolling a sum of 5 (X) Getting heads (H) Getting Tails(T) a number To get Probability of a sum of 5 Number of ways getting heads Expected value of tails in 5 tosses Definition - Expected value The sum of the outcomes of a random variable multiplied by its probability. Tossing a coin to get a head(h) five times, possible outcomes : 25 = 32 Number of heads(h) Probability (p) Product: Number of heads x probability hxp 0 1/32 0 x 1/32 = 0/32 1 5/32 1 X 5/32 = 5/32 2 10/32 2 x 10/32 = 20/32 3 10/32 3 x 10/32 = 30/32 4 5/32 4X 5/32 = 20/32 5 1/32 5 X 1/32 = 5/32 Total 80/32 Expected value = 0/32 + 5/32 + 20/32 + 30/32 + 20/32 + 5/32 = 0+5+20+30+20+5 32 = 80/32 or 2.5 Shortcut symbol to get the total or sum : 5 Expected value = ∑ hxp 0 ∑ 5 or simply ∑hp 0 =80/32 Examples of Random Variables Example: Tossing a coin: we could get Heads or Tails. Let's give them the values Heads=0 and Tails=1 and we have a Random Variable "X": In short: X = {0, 1} Note: We could have chosen Heads=100 and Tails=150 if we wanted! It is our choice. So: We have an experiment (such as tossing a coin) We give values to each event The set of values is a Random Variable Example: Throw a die once Random Variable X = "The score shown on the top face". X could be 1, 2, 3, 4, 5 or 6 3.06 Expectation and Variation Mean The average of the set of values of a random variable X. -b X = ∑𝑋 ----N Where ∑ 𝑋 means sum all the values of x n is the number of values. Example 1. A random variable X representing the occurrence of absences: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29 The sum of these numbers is 330 There are fifteen numbers. The mean is equal to 330 / 15 = 22 The mean of the above numbers is 22