Math 104 3 28 05 Probability These notes are a review of the material we covered in Chapters 13 through 18 1 Probabilities are always between 0 00 and 1 00 inclusive 2 Percentages are a way of expressing numbers between 0 00 and 1 00 For example 60 means the same as 0 60 and both mean the same as three fifths For calculations always convert percentages to numbers 3 Something NEVER happens its probability is 0 00 Something ALWAYS happens its probability is 1 00 4 The One minus trick The probability of an event NOT happening is equal to 1 00 minus the probability of its happening Exercise 4a If the probability of rolling two sixes with two dice is 1 36 then what is the probability of failing to roll two sixes Answer 35 36 that is 1 minus 1 36 Exercise 4b What is the probability of rolling two non sixes Non Answer Whatever it is it isn t related to the numbers in exercise 4a because two non sixes is NOT the opposite event to two sixes 5 One way to determine a probability is first to make a list of all the possible things that can happen Sometimes these are called all the possible outcomes of an experiment Construct your list so that all the outcomes are equally likely This requires common sense Then Probability of an event Number of ways the event can happen divided by Total number of things that can happen Exercise 5a Again what is the probability of rolling two non sixes Answer There are 36 equally likely outcomes when you roll two dice which could be written 11 12 13 56 66 If you write out the list and count the outcomes that have two non sixes you ll see that there are 25 of them So Probability of two non sixes 25 36 Exercise 5b If you spin a roulette wheel once what is the chance of getting 00 1 Answer There are 38 slots on the wheel and only one of them is labeled 00 so the probability is 1 38 6 Conditional probability By the conditional probability of an event given some condition we mean the probability you would calculate for the event on the assumption that the condition has already happened We write this P event condition or if we call the event B and the condition A we might just write P B A Exercise 6a Draw two cards from the top of a shuffled deck What is the conditional probability of drawing an Ace on the second draw given that you drew an Ace on the first draw Answer Assume that the condition ace on the first draw already happened Then there are 51 cards in the deck each equally likely to be drawn Three of them are aces So the conditional probability is 3 51 Exercise 6b What is the unconditional probability of getting an Ace on the second draw Answer In this case you know nothing about the first draw So there are 52 cards that are all equally likely to get drawn second Four of them are aces so the probability is 4 52 Exercise 6c Flip a fair coin ten times What is the conditional probability of getting a head on the tenth flip given that the first nine flips were all tails Answer 50 The tenth flip is independent of the first nine flips That s part of what we mean by a fair coin So no matter what you assume about the first nine flips the probability of getting a head on the tenth flip is just one half 7 Two events are independent if when one of them happens it doesn t change the probability of the other one happening If A and B are independent then P B A is the same as P B This is important in our chance process models the binomial model and the box model In these models we assume that the successive trials are independent of each other 8 The multiplication rule The probability of events A and B both happening is P A and B both happen P A happens times P B happens A happens The second factor is a conditional probability 2 Exercise 8a What is the probability of drawing two aces from the top of a shuffled deck Answer This can be described as A and B both happening if A means drawing an ace as the first card B means drawing an ace as the second card These events aren t independent So the answer is P A and B both happen P A happens times P B A which is 4 52 which is 3 51 So the answer is 4 52 times 3 51 about 0 0045 That s less than half of one percent 9 You can combine several events in the same way Exercise 9a What is the chance of getting four aces on the first four draws Answer P first ace times P second ace given that first was ace times P third ace given first two times P fourth ace given first three which is 4 52 that s 3 51 that s 2 50 that s 1 49 So the answer is 4 52 times 3 51 times 2 50 times 1 49 or about 0 0000055 That s pretty rare 10 The multiplication rule is easier when the events are independent because you don t have to think about conditional probabilities Exercise 10a Again what is the chance of rolling two non sixes Answer This is the chance of A and B happening if A non six on first die B non six on second die So the answer is P two non sixes P A that s 5 6 times P B A but that s just P B since B is independent of A So it s just 5 6 again 5 6 times 5 6 25 36 3 That s the same answer we got before But using the multiplication rule we didn t need to list all 36 outcomes Exercise 10b Consider a binomial model with n 4 trials and p 0 48 What is the chance of getting exactly 4 hits Answer In this case p means the probability of getting a hit on any one trial The trials are independent So the answer is P four hits P hit on first trial times P hit on second trial times P hit on third trial times P hit on fourth trial 0 48 times 0 48 times 0 48 times 0 48 about 0 053 or about five percent You would normally need the binomial formula for this kind of problem But when k the number of hits you need is the same as n the number of trials the multiplication rule is enough Exercise 10c Consider a binomial model with n 4 trials and p 0 48 What is the chance of getting exactly 0 hits Answer This is the chance of getting exactly 4 misses So the answer is P four misses P miss on first trial times …