Mathematics in Today's World

Last Time We talked about Cryptography

Today We will talk about probability. There are four rules that govern probabilities. One good way to analyze simple probabilities is to make a list of everything that can happen—this is called a sample space. A probability model is the sample space with probabilities for each outcome.

Random Phenomena A phenomenon is random if each individual occurrence is uncertain, but there is a pattern over many occurences. Examples of random phenomena: Tossing a coin, rolling a die, drawing a card. The gender of a baby.

Random Phenomena To describe random phenomena, start by giving a list of everything that could happen. Each distinct thing that could happen is called an outcome. If we toss a coin, there are two outcomes: heads or tails. We can’t predict which will happen, but in the long run 50% of the tosses are heads and 50% are tails.

Random Phenomena If we roll a (six-sided) die, there are six possible outcomes: We can’t predict in advance which outcome will occur. In the long run each outcome will occur 1/6th of the time.

Random Phenomena We don’t know in advance if a baby will be a boy or a girl. In the long run it turns out that about 49% of newborns are girls, and 51% are boys. Note that even though there are two outcomes, they don’t have to occur in equal proportions.

Random Phenomena To describe random phenomena, start by giving a list of everything that could happen. Each distinct thing that could happen is called an outcome. Each outcome of a random phenomenon has a probability. The probability of an outcome is the proportion of times it occurs over many trials. Probabilities are proportions, so we can express them as percents (50%), decimals (0.5), or fractions (1/2), whichever is most convenient. I will use all of these.

Random Phenomena We can restate our earlier observations in terms of probabilities: When we flip a coin the probability of getting tails is 50%. When we roll a die, the probability of getting a 4 is 1/6 The probability that a newborn baby is a girl is about 49%

Sample Space When there are not too many outcomes, one good way to analyze a random phenomenon is to make a list of every outcome. The set of all of the outcomes of a random phenomenon is called the sample space. For tossing a coin the sample space is {heads, tails} For rolling a die the sample space is {1, 2, 3, 4, 5, 6}

Random Phenomena We use the term event to describe a subset of the sample space. For example, when we roll a die, we can look for the event: “rolling a odd number.” There are three outcomes in this event: rolling a 1, a 3, or a 5. As a set, this event is {1, 3, 5} The probability of this event is 1 2

Random Phenomena Another event is “rolling a 6” There is only one outcome in this event. As a set, this it is {6} The probability of this event is 1 6

Probability models Heads Tails 0.50 The sample space tells us all of the possible outcomes. If we include the probability of each outcome we get what is called a probability model. Example Here is the probability model for tossing a coin: Heads Tails 0.50

Probability models Boy Girl 0.51 0.49 Here is a probability model for the gender of a newborn: Here is a probability model for tossing a coin: The top row is a list of the outcomes. The bottom row is the probability of each. Boy Girl 0.51 0.49 1 2 3 4 5 6 1/6

Probability Rules There are four rules which govern probabilities. Any probability is a number between 0 and 1. A probability of an event is the proportion of times that it can be expected to occur. The larger the probability, the more likely it is.

Probability Rules If the probability of an event is 1, it will always happen for sure. If the probability of an event is 0, it never happens. Note that neither of these are really random!

Probability Rules The probability that an event does not occur is 1 minus the probability that the event does occur. What is the probability that we roll a die and do not get a 6? The probability of getting a 6 is 1 6 . By the rule, the probability of not getting a 6 should be 1− 1 6 = 5 6

Probability Rules If two events have no outcomes in common, they are said to be disjoint. The probability that one or the other of two disjoint events occurs is the sum of their individual probabilities. The two events “rolling an odd number” and “rolling a 6” are the sets {1, 3, 5} {6} These sets are disjoint (nothing in common).

Probability Rules By the rule, the probability that we roll either an odd number or a 6 is the sum of their individual probabilities: 1 2 + 1 6 To add these fractions we can use a common denominator of 6 3 6 + 1 6 = 4 6 This can be reduced 4 6 = 2 3

Probability Rules All possible outcomes together must have probability 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly one.

Probability models 0.30+0.20+0.20+0.10+0.10+Blue =1 We can use this last rule to fill in missing information in a probability model. If you draw an M&M candy at random from the bag, it will have one of six colors. Here is a probability model: What is the probability of randomly drawing a blue M&M? The probabilities of all the outcomes must add up to 1. So 0.30+0.20+0.20+0.10+0.10+Blue =1 The probability the M&M is blue is 0.10 Brown Red Yellow Green Orange Blue 0.30 0.20 0.10 ?

Independent Events If the outcome of one event has no impact on the outcome of the other, we say the two events are independent. Each roll of a die is independent from any other roll. Why? Because the outcome of one roll does not affect the next roll. The weather one day is not independent from the weather the next day. Why? Because weather moves in systems: if it rains one day, the probability of it raining the next day is increased.

Independent Events The most important fact about independent events is that the probability that both happen is the product of the individual probabilities. The probability that I get heads on a coin toss is 1/2. The probability that I get a 1 when I roll a die is 1/6. These are independent events. So if I toss a coin then roll a die, the probability that I get heads on the coin and a 1 on the die is the product: 1 2 ⋅ 1 6 = 1 12

Probability models We are now going to put all of this information together to start constructing more complicated probability models. Remember a probability model is a list of outcomes and their probabilities.

Probability models If we toss two coins, what are the possible outcomes? We can get heads on both, tails on both, or heads on one and tails on the other. But be careful, there are actually four outcomes. Why? This is easiest to see if you imagine the coins being different, say one is a quarter and one is a dime: Quarter Dime H T

Probability models Now we have a sample space with four outcomes. Let’s make a probability model. What is the probability of getting heads on each coin? The coin tosses are independent, so probability of heads on the quarter and heads on the dime = (probability of heads on the quarter) X (probability of heads on the dime) The probability of getting heads on both is 0.50 0.50 =0.25 In a similar way the probability of each of the other three outcomes is also 0.25

Probability models So the probability model for tossing two coins is: What is the probability of tossing two coins and getting one head and one tail? This is an event, containing two outcomes: T,H and H,T So the probability of one head and one tail is 0.25+0.25=0.50 H,H T,H H,T T,T 0.25

Probability models Here’s a related example: let’s find a probability model for rolling two dice. First we should make a sample space: Again we distinguish getting a 3 on the first die and a 4 on the second from getting a 4 on the first and a 3 on the second.

Probability models It is useful to write outcomes here as pairs of numbers. So (3,4) indicates a 3 on the first die and a 4 on the second. As I said on the last slide: the outcome (3,4) is not the same as the outcome (4,3). Now we know the sample space. What are the probabilities of each outcome?

Probability models It turns out that each outcome has the same probability: How many outcomes are there? 36. They all have the same probability, so each one has probability 1/36.

Probability models Now, suppose we roll 2 dice and then add the numbers we get. So if we get (2,5), the outcome is the sum of these 2+5=7 What is the sample space? The possible outcomes (sums of two numbers between 1 and 6) are {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Probability models What’s the probability of each of these outcomes? Look back at the sample space for rolling two dice: We get a sum of 2 only from the outcome (1,1). The probability is 1 36 There are two ways to get a sum of 3: (1,2) and (2,1). The probability is 1 36 + 1 36 = 2 36 = 1 18

Probability models How many ways are there to get a sum of 4? There are three ways: (1,3), (2,2), or (3,1). The probability of getting a sum of 4 is: 1 36 + 1 36 + 1 36 = 3 36 = 1 12 Continuing in this way, we get the following probability model: Outcome 2 3 4 5 6 7 8 9 10 11 12 Probability 1 36 2 36 = 1 18 3 36 = 1 12 4 36 = 1 9 5 36 6 36 = 1 6

Probability models It can be useful to represent probability models using probability histograms. The horizontal scale gives the outcomes, and the probability of each outcome determines the height of the bar: