Probability And Expected Value ———————————— Picture source: http://www.sxc.hu/photo/732677

Probability –———————— Probability is the measure of how likely to occur an event is. To calculate the probability, use the following formula: the number of ways an event can occur the total number of possible outcomes P =

For Example –———————— There are 26 letters in the alphabet. Five of the letters are vowels. So the probability of choosing a vowel from the alphabet is: 5 26 = the number of vowels total letters in alphabet P(vowels) = P(odd)= 6/11

P = 6 ÷ 11 Practice –———————— A number from 1 to 11 is chosen at random. What is the probability of choosing an odd number? Remember our formula: the number of ways an event can occur the total number of possible outcomes P = Hint: There are 6 odd numbers and there are 11 numbers we could choose from. P = 6 ÷ 11

Independent events —————— If there is a jar with 10 maroon marbles and 10 white marbles in it, what is the probability you’ll choose a maroon marble? On your first try you drew a maroon marble. If you then replaced the maroon marble and drew again, what’s the probability that you will draw another maroon marble? P(maroon) = 10/20 = ½ P(maroon second try) = 10/20 = 1/2 These are called independent events since the two events (your first draw and your second) do not affect one another.

Dependent events —————— If, on your first try, you drew a maroon marble, kept it, and then drew a second marble, what’s the probability that your second marble is also maroon? **Hint: the total number of marbles has changed!** What about the probability for drawing a white marble on your second try? P(maroon second try)= 9/19 P(white second try) = 10/19 These are called dependent events. The probability of your second draw is dependent on what you drew the first time.

Lotteries –———————— In lotteries you must pick the correct numbers to win, but there are different versions of the game. Version 1: Order does NOT matter, numbers CANNOT repeat Suppose that to win the lottery we must choose any three numbers 0-9 in any order. The numbers cannot repeat. Let’s say we chose the numbers 6,9, and 4. As long as our three numbers are drawn in any order, we win! To find the probability that we will win, we have to multiply the probabilities that each of our numbers will be chosen.

Lotteries –———————— 1 2 3 4 5 6 7 8 9 On the first draw the probability that a 6,9, or 4 will be drawn is . Out of 10 possible numbers, 3 numbers can be chosen. Let’s pretend a 4 was drawn the first time. On the second draw the probability that a 6 or a 9 will be drawn is . Because we can’t repeat, the 4 that we drew on the first draw has to be excluded from this second draw making our total possible numbers 9. The numerator is now 2 because there are only two number choices that can occupy this space to make us winners. A 6 was drawn on the second draw. The probability that our last number, 9, will be chosen on the third draw is . We have already drawn two numbers so our total choices are now 8. There’s only one number left that can be chosen that will make us a winner. winning numbers

Lotteries –———————— 1 2 3 5 7 8 9 We now multiply all the individual probabilities together to determine the probability that all three of our numbers will be chosen and we will win. We can expect to win only 1 out of every 120 lotteries!! winning numbers 4 6

7 5 1 8 Lotteries –———————— 3 Version 2: Order DOES matter, numbers CAN repeat Again, we can choose any number 0-9. The order that we put these numbers in must match the order in which the lottery numbers are drawn. The numbers can repeat. Let’s say we chose the numbers 2,8, and 2 again in that order. 9 6 On the first draw, the probability that a 2 will be drawn out of a total of 10 available numbers is . 2 Because our numbers can repeat, we have to replace the 2 so that our total possible numbers to draw from is again 10. 4 The probability that we will now draw an 8 is . Again, we replace the number we drew. The probability that our final number will be a 2 is 1 out of 10 .

Lotteries –———————— We now multiply the probabilities together to determine our overall probability of winning: Which lottery version would you rather play? The one where order doesn’t matter and numbers cannot be repeated or the one where order does matter and repeating is okay? Why? The version where order doesn’t matter and repeating is not okay is better because your chances of winning are greater. Order matters, can’t repeat: 1/10 x 1/9 x 1/8 = 1/720 You cannot replace the number you drew so the total number of choices decreases by one for each draw.

Practice –———————— The answer is: 1/720 How would you calculate the probability that you would win if the version of the lottery game had rules that order mattered and you were NOT allowed to repeat your number? The answer is: 1/720

Expected Value –————— Expected Value is defined as the sum of all possible values for a random variable, each value multiplied by its probability of occurrence. For example, in the game Hog, a player rolls a dice. If the dice lands on a 1, the player gets zero points. If it lands on any other number, the player gets one point for every dot on the dice. So if I rolled a 4, I get four points. To calculate the expected value of rolling a dice in Hog, you multiply the amount you win by the probability of rolling a specific number. Each number on the dice has a 1/6 probability of being rolled. Calculate this for each potential outcome and add them all together. http://www.amstat.org/publications/jse/v11n2/02-006R2_Feldman.doc This means that if you roll the dice 10 times, your expected score will be about 33!

If you play ten times, you Expected Value –————— Pretend I flip a coin. If it lands on heads you win $5. If it lands on tails, you win nothing. What is the expected value? I flip the same coin. This time if it lands on heads you win $5, but if it lands on tails, you win $3. What is the expected value? This time, you win $5 if the coin lands on heads, but if it lands on tails, you owe me $6. Calculate the expected value. If you play ten times, you Can expect to lose $5.

Wrap it up –———————— Remember some odds depend on what already happened, some odds don’t Remember the odds of winning are important But, the expected value is even more important