15.301/310, Managerial Psychology Prof. Dan Ariely Recitation 3: Probability (a dash “-“ indicates student responses) Probability: An introduction Today we talk about probability theory It is the first step in understanding statistics. Probability theory – a systematic way to think about outcomes in a stochiastic world What do we know about probability? -probabilities are not certain Why is the world not certain? If you go to someone in the morning, you cannot be 100% sure what they will say back. Why not? Why do people react in different ways to different things? -because they have the choice If we have a pile of bricks and we kick them, they always do the same thing If we have people and we kick them…. They react differently 1. The world is really probabilistic 2. We might not be able to take everything into account. If could write an equation with everything included, the world might be deterministic But as we see it, it’s probabilistic. We use intuition all the time When you study for exams, you succeed or not It is not always clear what will happen What is the probability of a newborn being a male? ~50% HH in coin toss .25 Get a 6 on a die? 1/6 Choosing the one blue ball out of a bag of 5 balls 1/5, one of the others, 4/5 Probability and Frequency – Binomial distributions Systems with 2 possible outcomes Like: Die or stay alive Sick or healthy Yes or no__ __ __ __ __ __ __ You don’t always have to ask about one event. What is the probability that out of 100 kids, 17 will be girls, and the rest boys… Or first x girls, then y boys, then z girls, etc…. Possible outcomes of tossing 2 coins: HH HT TT What are the probabilities of getting each? (.25 .5 .25) Think of 3 coins What are the possible outcomes? (HHH HTT HHT TTT) Please write down the probabilities for each of these events. (1/8 3/8 3/8 1/8) For 6 coins? (if we don’t care about order?) How many outcomes? (7) Please write down the probability for each of these 7. HHHHHH THHHHH TTHHHH TTTHHH TTTTHH TTTTTH TTTTTT How do you do it? -the easiest way is the binomial solution Suppose we want to enumerate it, how do we do it? There are 2 components if we calculate by hand: how many possible outcomes? How many times would they occur? Multiply the outcomes by how many times they occur. We just have to figure out all the combinations. It gets harder and harder for larger amounts. We have to come up with something better. Look at these 3 distributions (see slide). What can you learn from the differences of these 3? -more trials you get, the more it looks like a bell shaped curve -symmetrical -more possibilities gives a lower probability overall (for individual events) If I threw an infinite number of coins, what is the probability that I will have half and half, heads and tails? -Exactly 0. If there are infinite coins, you get a slimming bar corresponding to 50/50, slims to 0 as the coins go to infinity The probability of any particular event goes to zero because you divide by infinity 15.301/310, Managerial Psychology Recitation 3 Prof. Dan Ariely Page 2 of 10If we throw 2 times or 6 times, which is the better chance of getting a 50/50 distribution? -2 times. There are more possibilities for outcomes if we toss 6 times. Fewer trials means more width devoted to each option. When we toss 2 coins, the probability of getting 50/50 is .5. At 6 coins, it is .3-something It is counterintuitive to get farther away as we toss more coins. We would expect that the more coins we throw, the closer we would get to 50/50. How do we solve this problem? -going discreet to continuous It is important to focus on the range, not a specific value. Probability of .5 boys is less the bigger the hospital is. We need to take the probability .5 or higher to get higher precision Or if we define the distribution from .5, define a range, a width, an interval. Other binomial distributions Questions about Class We had some questions from students The basic question – what exactly is this class? And there were many related questions. Let me give general answer: This class is technically known as a mish mash. Over the years it became communication intensive – the writing, presentations. And there is also the lab part – we cover statistics, etc. Then the rest is managerial psychology So how do you cover managerial psychology in 8, 9, 10 lectures? I could have chosen a topic, lets say labor, and concentrated on that. But your path is not clear to me, probably not to you either. So we are doing an overview and I will give you an idea of how to apply what you are learning to management Lets talk about the lecture on perception – how does that relate to management? For example, Adaptation – how does adaptation link to salaries? If you have a pay level, you get used to it, and you are happy if there is a change and you get a raise. Think about the basic principles and apply them to different scenarios I don’t want to tell you how to apply them to what is interesting to you. Start with perception and memory, go from there We will get more and more applied as we go through the topics. Relativity of perception is one of the central themes. Addressing concerns about exams: 15.301/310, Managerial Psychology Recitation 3 Prof. Dan Ariely Page 3 of 10-10 We will give you practice questions before the exam. Basically, you have to do the reading. Either think about measurement or managerial psychology Take the alien abduction article last week – what mechanisms in the paper caused people to think they were abducted? What is the main message? We are not going to ask when the article was published, etc. Are there other questions about the structure of the class? No? Fabulous. Now we will talk about binomial distribution Lets look at a 60/40 distribution. (Slide) There is a probability of .6 that people will die in a situation, what is the likelihood 10/10 survive? -They are independent events, multiply them. It doesn’t make sense to add, subtract, divide. Multiplication is the only operation I could think of that made sense. Multiply probabilities of each because they are independent -> .410 What is the likelihood 9/10 will survive? - .49 times .6 Take the probability of 9 people surviving, and multiply by .6, is that right? We could also read it as 9/10 or 10/10 survive – at least 9 people survive. What do we do between the two?