Basic Probability

Let ! = #|8 ≤ # < 13, # ∈ ℕ-, ., and / are the subsets of ! such that- = multiples of four. = factors of 24/ = square numbers

(a) List the elements of !.

(b) (i) Draw a Venn diagram to show the relationship between sets -, ., and /.

(ii) Write the elements of ! in the appropriate places on the Venn diagram.

(c) List the elements of:

(i)- ∩ / (iI) -′ ∩ . ∩ /

(d) Describe in words the set - ∪ ..

(e) Shade in the region on you Venn diagram that represents - ∪ / ∩ .′.

Sample Space & Basic Probability

¨ The probability of an event ! is the number of outcomes in the even out of the total possible outcomes in the sample space, ".

# ! = numberofoutcomesin!totalnumberofoutcomes

¨ The sample space contains all the possible outcomes of an event. ¨ For example, if a coin is flipped, the sample space would be heads (5) or tails (7).

Sample Space & Basic Probability

Some important facts about probability:¨ An event that is certain to happen every time has a probability of one.

This is the largest probability possible.¨ For example, the probability of rolling a die and landing on a number

is one. There is no other option when a die is rolled other than landing on a number.

Sample Space & Basic Probability

Some important facts about probability:¨ An event that will never happen has a probability of 0. This is the

smallest probability possible.¨ For example, the probability of rolling a number cube and landing on

the letter A is 0. It is impossible to roll a number cube and land on a letter.

Sample Space & Basic Probability

Some important facts about probability:¨ The probability of event ! is 0 ≤ $ ! ≤ 1, meaning that the

probability is any number between 0 and 1 inclusive.¨ Probabilities can be written as fractions, decimals, or as a percent.

Sample Space & Basic Probability

Some important facts about probability:¨ The complement of any event ! would be all the outcomes in the

sample space that are not in !. This is very similar to the complement of a set.

¨ For example, if we flip a coin and look at landing on tails, then the complement of the event is landing on heads. So " # + " #′ = 1. This says that the probability of landing on tails and the probability of not landing on tails is one.

¨ The IB Math Studies formula packet has complementary events as " !′ = 1 − " ! .

Sample Space & Basic Probability

Joe has a bag containing 5 red marbles, 7 blue marbles, and 3 yellow marbles. He reaches in and randomly selects one marble.

What is the probability Joe selects:1. a red marble?

! " = 515 =

13 or0.333

2. a yellow marble?

! , = 315 =

15 or0.2

3. a marble that is not blue?

! .′ = 815 or0.533

Sample Space & Basic Probability

¨ In a set number of probability trials, the expected value represents how many times you would expect a certain event to occur.

¨ For example, suppose you flip an unfair coin where landing on tails has a probability of 0.75. If you flip the coin 100 times, you would expect 75 of the flips to be tails.

Sample Space & Basic Probability

Ava-Taylor is interested in what students are eating for lunch in the school cafeteria. For one week she records the choices of 100 students. The results are displayed in the table below.4. Use the given information to find the probability that a student chosen at random:(a) selected pizza

21100 or0.21

(b) selected fresh fruit20100 =

15 or0.2

(c) did not select a hot dog84100 =

2125 or0.84

Sample Space & Basic Probability

5. If 350 students are randomly sampled, how many students would you expect to select French fries?

350 28100 = 98

98 students

Mutually Exclusive, Combined, & Independent Events

¨ Mutually exclusive means two or more probability events have no common outcome. This means ! " ∩ $ = 0.

¨ For example, roll a die and focus on the events of rolling a number less than 4 or a number greater than 5. This can be expressed as ! numberlessthan4orgreaterthan5 .

¨ These two outcomes have no common outcome, so they are mutually exclusive.

¨ A number cannot be less than 4, while also greater than 5.

Mutually Exclusive, Combined, & Independent Events

¨ For example, roll a die and focus on the events of rolling a number less than 4 or a number greater than 5. This can be expressed as ! numberlessthan4orgreaterthan5 .

¨ How can we show the event using a Venn diagram?¨ To find the probability of the event, add the individual probabilities

together.! numberlessthan4orgreaterthan5= ! numberlessthan4 + ! greaterthan5

Mutually Exclusive, Combined, & Independent Events

¨ If two or more events have a common outcome, they are considered combined events.

¨ Using the example of rolling a die, focus on the events rolling an even number or rolling a number less than 5. This can be expressed as ! anevennumberoranumberlessthan5 .

¨ How can we show the event using a Venn diagram?¨ To IB Math Studies formula booklet has the formula

! 1 ∪ 3 = ! 1 + ! 3 − ! 1 ∩ 3for combined events.

Mutually Exclusive, Combined, & Independent Events

¨ Independent events are events that happen together but one does not affect the outcome of the other.

¨ For example, if a coin is flipped twice, it does not matter if the coin lands on heads the first time because the second flip is not affected by the first flip.

¨ The IB Math Studies formula booklet has the following formula for independent events ! " ∩ $ = ! " & ! $

Mutually Exclusive, Combined, & Independent Events

¨ If Taylor has a 0.45 probability of winning a 100m race, then determine the probability she would win two races, assuming the outcomes are independent.

0.45 0.45 = 0.2025

Basic Probability

Academic Agenda:¨ Complete the Basic Probability Assignment.¨ Closing Questions – 17 & 18

An afterschool club has 80 students. On Friday, the students are offered three snack items: fruit ("), crackers ($), or granola (%).

¨ Homework: Review class notes and finish the Basic Probability Assignment.