Simple Probability

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.

What is the probability that a card drawn at random from a deck of cards will be an ace? Since of the 52 cards in the deck, 4 are aces, the probability is 4/52 or 1/13.

The same principle can be applied to the problem of determining the probability of obtaining different totals from a pair of dice. As shown below, there are 36 possible outcomes when a pair of dice is thrown.

What is the probability of throwing a total greater than 8?

Calculate the probability of fewer than 5 sales.

What is the probability of at least one sale?

Calculate the probability of 5 to 9 sales inclusive.

A biased die has the following probabilities of facing upwards.

Which number is most likely to be facing upwards after this die is tossed?

Calculate the value of k. 0.05

What is the probability that the number facing up is odd? 0.55

What is the probability that the number facing up is even or a multiple of 3? 0.60

Exercise 1.1

Assume that there are n people in the room. Ignoring leap years, what is the probability that no one else in the room shares your birthday?

Assume that there are 253 people in the room. Ignoring leap years, what is the probability that no one else in the room shares your birthday? What is the probability that someone else in the room shares your birthday?

Assume that there are n people in the room. Ignoring leap years, what value of n (most closely) makes the probability that someone else shares your birthday (1/n)?

Graphics calculator n = 19

Assume that there are n people in the room. Ignoring leap years, what value of n (most closely) makes the probability that two people in the room share birthdays equal to 0.5?

Assume that there are n people in the room. What value of n (most closely) makes the probability that two people in the room were born on the same day of the week equal to 0.5?

Do you think this is true?

Are we making some assumptions when we make these calculations?

What assumption are we making?

Is it a fair assumption?

Venn Diagrams

R

R’R’

R + R’= 1

P(R)+ P(R’)= 1

Combining

AND

OR

?

?

?

?

AND

If R and G are independent

OR

Write this in another way.

THREE SETS: RED, GREEN, BLUE

Write down what this area is.

Consider the numbers

R = MULTIPLES OF 2 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

G = MULTIPLES OF 3 3, 6, 9, 12, 15, 18,

Organize 3, 9, 15 2, 4 8,10, 14,16, 20 6, 12, 18

What’s missing? 3, 9, 15 2, 4 8,10, 14,16, 20 6, 12, 18

What’s missing? 3, 9, 15 2, 4 8,10, 14,16, 20 6, 12, 18 1, 5, 7, 11, 13, 17, 19

Find the probabilities.

Independent?

P(R)=?

P(G)=? 0.15

Does

Example p. 5 Probability of avocado = 0.75 Probability of bacon = 0.80 Probability of both = 0.65 Write these values in symbols

Example p. 5 Probability of avocado = 0.75 Probability of bacon = 0.80 Probability of both = 0.65 Draw the Venn diagram

Example p. 5 Probability of avocado = 0.75 Probability of bacon = 0.80 Probability of both = 0.65 Calculate the probability of neither Using Venn diagram Using equation

A B

Mutually Exclusive

16 girls in a class 7 hockey players 9 softball players 4 play both How many play neither sport?

345 4

24 boys live in Manukau Rd 13 soccer players 12 cricket players 7 play both How many play neither sport?

S C

24 people attend a party 4 eat pizza but not chicken 6 eat neither chicken nor pizza 7 eat both How many people eat chicken but not pizza?

P C

30 boys in a class All play rugby or basketball 19 play rugby 15 play basketball How many play both games?

R B

23 girls in a class 13 can type 12 take woodwork 7 do neither How many can type and take woodwork?

T W

30 men 17 use Pong after- shave 19 use Macho after- shave 4 use neither How many use both?

P M

A women who is bright, single, 31 years old, outspoken and concerned with issues of social justice is most likely to be a.A bank teller b.A bank teller and a feminist

Over the course of a year, for which type of hospital would you expect there to be more days on which at least two thirds of the babies born were boys. a.a large hospital b.a small hospital c.it makes no difference

In a family of six children the sequence BGGBGB is__?___as BBBBGB a.more likely b.less likely c.the same as

The average score for all secondary students in a district is known to be 400. You pick a random sample of 10 students. The first student you pick had a score of 250. What would you expect the average to be for the other 9? a.more than 400 b.less than 400 c.400

The average score for all secondary students in a district is known to be 400. You pick a random sample of 10 students. The first student you pick had a score of 250. What would you expect the average to be for the entire sample of 10? a.more than 400 b.less than 400 c.400

Three girls have respective probabilities of 0.8, 0.6 and 0.7 of independently passing an examination. Find the probability that All pass None pass At least one passes Just one passes

Which is more likely? A man a.had a heart attack and is over 55, b. had a heart attack given that he is over 55.

Exercise 1.2 and 1.3

Here's a puzzler for you all: You and two of your friends get into a dispute and decide to solve it with a "truel", a three way duel. Friend #1 is a crack shot, never missing his target. Friend #2 hits his target 2/3 of the time. You hit your target 1/3 of the time.

The truel It is decided that you will take the first shot, the 2/3 marksman will take the second shot (if still alive) and the 100% marksman will go last. This will continue until there is only one left alive.

The ‘truel’ On your turn you get to fire one bullet. You get to go first. In order to maximize your chances of living thru this, where should you take your opening shot? And what are your chances of winning the truel if you follow this strategy?

Duck Hunting! Ten duck hunters are all perfect shots 10 ducks fly over. All 10 hunters pick a duck at random to shoot at, all 10 hunters fire at the same time.

Duck Hunting! How many ducks could be expected to escape, on average, if this experiment were repeated a large number of times?

Probability Trees

Conditional probability

Definition A conditional probability is the probability of an event given that another event has occurred.

For example, what is the probability that the total of two dice will be greater than 8 given that the first die is a 6? There are 6 outcomes for which the first die is a 6, and of these, there are four that total more than 8

If A and B are two events then the conditional probability of A given B is

Independence of Events

Definition In probability theory, two events are independent if the occurrence of one is unrelated to the probability of the occurrence of the other. Getting heads the second time a fair coin is tossed is independent of getting heads on the first toss. There is simply no valid way to predict the second outcome from knowledge of the first.

A and B are two events. If A and B are independent, then the probability that events A and B both occur is: p(A and B) = p(A) x p(B). In other words, the probability of A and B both occurring is the product of the probability of A and the probability of B.

If A and B are Not Independent If A and B are not independent, then the probability of A and B is p(A and B) = p(A) x p(B|A) where p(B|A) is the conditional probability of B given A.

Mutually Exclusive

Definition If events A and B are mutually exclusive, then the probability of A or B is simply: p(A or B) = p(A) + p(B).

Probability of A or B

Consider the probability of rolling a die twice and getting a 6 on at least one of the rolls. The events are defined in the following way : Event A: 6 on the first roll: p(A) = 1/6 Event B: 6 on the second roll: p(B) = 1/6 p(A and B) = 1/6 x 1/6 p(A or B) = 1/6 + 1/6 - 1/6 x 1/6 = 11/36

Alternate approach The probability of getting a number from 1 to 5 on the first roll is 5/6. Likewise, the probability of getting a number from 1 to 5 on the second roll is 5/6. Therefore, the probability of getting a number from 1 to 5 on both rolls is: 5/6 x 5/6 = 25/36. This means that the probability of not getting a 1 to 5 on both rolls (getting a 6 on at least one roll) is: 1-25/36 = 11/36.

Despite the convoluted nature of this method, it has the advantage of being easy to generalize to three or more events. For example, the probability of rolling a die three times and getting a six on at least one of the three rolls is: 1 - 5/6 x 5/6 x 5/6 = 0.421