probability

Question 1

probability

Answer 1

• at least 3 senses of probability
• these different senses are often employed in different contexts, because some make more sense in some contexts relative to others
• three of them are: the classical view of probability, the frequency view of probability, the subjective view of probability

Question 2

the classical view

Answer 2

• it is often used in the context of games of chance like roulette and lotteries
• the classical view - if we have an (exhaustive) list of events that can be produced by some (exhaustive) list of equiprobable outcomes (the number of events and outcomes need not be the same), the probability of a particular event occurring is just the proportion of outcomes that produce that event
• if we’re interested in a particular event then we just count the number of outcomes that produce that event
• if you’re viewing probability like this, its very important to be clear about what counts as a possible outcome

Question 3

the frequency view

Answer 3

• when you take a frequency view of probability you’re making a claim about how often, over some long period of time some event occurs
• the frequency view is often the view that we take in science. If we wanted to assign a probability to the claim “drug x lowers depression”, we cant just think of each possible outcome that could occur when people take drug x and then count up how many lead to lower depression and how many do not.
• no way to make an exhaustive list of every possible outcome
• but we can run an experiment where we give drug x and see whether it lowers depression, and we can repeat this many times, then we count up the proportion of experiments in which depression was lowered
• that is then the probability that drug x lowers depression

Question 4

the subjective view (credences)

Answer 4

• Consider the following statement - the Australian cricket team will lose the upcoming test series against south Africa.
• There is a sense in which you can assign a probability to this.
• But it isn’t the classical kind—we can’t just enumerate all the possible outcomes that lead to this event
• Nor is it the frequency kind—we can’t repeat the 2022/2023 cricket tour over and over and see how often Australia lose.
• When we talk about probability in this context mean something like a degree of belief, credence, or subjective probability.
• Probability in this context is the answer to the question “how sure are you that the Australian cricket team will lose the upcoming test series against South Africa?”

Question 5

calculating with probability

Answer 5

• the different views of probability have got to do with what the numbers mean, but once we have the numbers there’s no real disagreements about how we do calculations with those numbers
• some properties will help us to do calculations
• when we attach numbers to probabilities those numbers must range from 0 to 1
• if an event has probability 0 then it is impossible
• if an event has probability 1 then it is guaranteed
• these two simple rules can help us to check our calculations with probabilities, if we get a value more than 1 or a value less than 0, then something has gone wrong

Question 6

the addition law

Answer 6

• whenever two events are mutually exclusive - the probability that at least one of them occurs is the sum of their individual probabilities
• If we flip a coin, one of two things can happen. It can land Heads, or it can land Tails. It can’t land heads and tails (mutually exclusive), and one of those things must happen (it’s a list of all possible events).
• What’s the probability that at least one of those events happens? Since one of those events must happen the probability must be 1.
• But we can work it out from the individual probabilities.
  
  1. 1/2 possible outcomes produces Heads - P(Heads) = 0.50
  2. 1/2 possible outcomes produces Tails - P(Tails) = 0.50
• The probabilities of at least one of Heads or Tails occurring 0.5 + 0.5 = 1

Question 7

mutually exclusive and non-mutually exclusive events

Answer 7

• Consider a deck of cards:
  
  1. What is the probability pulling out a spade or a club?
  2. What is the probability of pulling out a spade or an ace?
• In situation 1 the events are mutually exclusively or disjoint. A card can’t be a spade and a club. It will either be a space, a club, or something else.
• The addition rule applies:
  
  • P(spade) + P(club) = probability of selecting a spade or a club.
• In situation 2 the events are not mutually exclusive. A card out be both a spade and an ace
• So we need a different rules.

Question 8

not double counting

Answer 8

• The red circles with white dots get counted twice
• So we need to subtract this amount
• First we work out P(red) + P(dot)
• Using the numbers on the previous slide this gives us 20/20
• Then we subtract 5/20
• This gives us P(Red U Dot) = 15/20
• But if all moths is too difficult, then we can just work out the probability by counting. All we need to do is to count up all the circles that are either red or have a dot. And we just divide that by the total number of circles.

Question 9

two or more events

Answer 9

In that last example we’re dealing single event where we could, for example, select:
1. A circle that is red
2. A circle with a white dot
3. A circle that is blue
- But sometimes we have to deal with multiple events
- We’ve already seen an example with coin flipping.
- Let’s say we flip a coin three times, we might want to work out the probability of getting, for example, heads, then tails, and then heads again.
- We can’t just add up the probabilities, before we work it out mathematically, we’ll work it out by counting
- If you don’t want to count, and you just want to work it out mathematically, then you can do this by just multiplying together the probability for each of the events

Question 10

independence and non-independence

Answer 10

• In the previous example, the two choices were independent
• This means that knowing whether you got heads/tails on the first flip didn’t impact how you calculated the probability of getting heads/tails on the second flip.
• We can calculate the probability of each event without considering anything about the other event.
• But sometimes this isn’t the case…sometimes knowing what happened on the first event changes how to calculate the probability of for the second event.

Question 11

conditional probability

Answer 11

• Let’s say we’re going to roll a dice
• But instead of just rolling a die, we’ll first select one of two dice. The set up is as follows:
  
  1. First, pick either a 20-sided dice or a 6-sided dice
  2. Second,, roll the dice
• If I ask you, what is the probability of rolling a 20 then answer you give will change if I tell you what die you picked.
• For the coin flip, being told about the first flip doesn’t change your calculation for the second.
• For the dice roll, being told you picked the D-20 or D-6 does change your calculation.
• If you picked a D-20 then the probability you rolled as 20 is 1/20
• If you picked a D-6 then the probability that you rolled a 20 is 0

Question 12

working with conditional probabilities

Answer 12

• We often encounter conditional probabilities in every day life where we use some bit of information to help us work out the probability of something.
• However, reasoning about conditional probabilities can be difficult and as a result people make a lot of mistakes when dealing with them.
• The most common mistake that you’ll encounter is the confusion being P(A|B) and P(B|A)
• Or as in the dice example:
  
  • P(roll 20|D20), which is 1/20
  • P(D20|roll 20), which is 1
• The other common mistake is confusing the conditional probabilities for the unconditional probabilities
• That is, confusing, for example P(A|B) and P(A)
• Or in the dice example:
  
  • P(D20|roll 20), which is 1
  • P(D20) which is 0.5
• There is a mathematically formula that relates all these quantities together.
• This is known as Bayes theorem
• Bayes theorem - allows us to update our probability calculations when we find out new information.
• E.g., we can update our calculation for rolling a 20 when we find out that we selected a D-20.

Question 13

bayes theorem

Answer 13

• Bayes theorem can help us think through conditional probabilities, because sometimes conditional probabilities can be very unintuitive
• Consider the following example - does a positive test mean somebody is actually sick?
• There is a test for an illness. The test has the following properties:
  
  1. About 80% of people that actually have the illness will test positive
  2. Only -5% of people that don’t have the illness will test positive
• Somebody, who may be sick or healthy, takes the test and tests positive
• Is that person actually sick?