MDM 4UIUnit 3: Probability

Day 2: Theoretical Probability

Theoretical Probability: Theoretical Probability

where P(A) is the probability that event A can occur, n(A) is the number of ways it can occur, and n(S) is the total number of possible outcomes in the sample space.

Probability that is calculated by counting all possible outcomes of an event in a sample set. This is sometimes called “classical probability”.

Example 1: Calculate Theoretical ProbabilityA bag contains five red marbles, three blue marbles, and two white marbles. What is the probability of drawing a blue marble?

∴ The probability of drawing a blue marble is 30%.

Example 2: Calculate Theoretical ProbabilityIf a single die is rolled, determine the probability of rolling (a) an even number (b) a number greater than 2.a) Even numbers are 2, 4 or 6.

∴ The probability of rolling an even number is 50%.

b) Greater than 2 are 3, 4, 5, 6

∴ The probability of rolling greater than 2 is about 66.7%.

Try on your own….Lee has a green hat, a black hat, and a grey hat. She also has green gloves, black gloves, and red gloves. In a hurry, Lee randomly grabs a hat and gloves. Determine the probability that the hat and gloves are the same colour.You may use any counting method that we have covered so far. This includes diagrams, charts, etc.There are 3 ways to choose a hat, then 3 ways to choose the gloves. There are only two possibilities where they are the same colour (both green or both black). ∴ The probability is about

22.2%.

Example 3: Calculate Theoretical ProbabilityFiona has two shirts (one red and one blue) and three pairs of pants (one green, one white, and one orange)that she can wear to her co-op placement.a) Suppose Fiona randomly picks a shirt and a pair of pants . Draw a tree diagram to illustrate Fiona choosing an outfit.

R

B

GWOGWO

RGRWROBGBWBO

b) Identify the sample space for this probability experiment. c) Fiona does not like to wear blue and green together, nor does she like orange and red together. What is the theoretical probability that she will randomly pick a shirt and pants combination that she does not like?

The sample space is all possible outcomes of Shirt-Pants.Red-Green, Red-White, Red-Orange, Blue-Green, Blue-White, Blue-Orange.

∴ The probability is about 33.3%.

d) Suppose Fiona wants to decrease the probability of drawing a combination she does not like. If Fiona buys another pair of pants, which colour should she choose – green, white, or orange – and why?

In order to decrease the probability, you need to increase the size of the sample space without adding to the event space.

If she choose green or orange, she will add to both the sample and event spaces.

To ensure that the probability decreases, she should purchase a white pair….that will only increase the sample space.

ComplementThe complement is the set of possible outcomes not included in an event.If one event is A, then the event A′ is all of the possible outcomes not in A. A′ is read “A complement”.

This relationship can be rearranged into two other useful forms:

Example 4: Probability of a ComplementBattleship is a game in which two opponents use a coordinate grid to try to guess the location of each other’s ships. Examine the board shown. Suppose a location is guessed at random.a) What is the theoretical probability of

hitting a ship on the first guess?

∴ The probability is 17%.

b) What is the probability of a miss on the first guess?

Try on your own….A box contains 3 bran, 4 banana, 5 blueberry, and 3 carrot muffins. What is the theoretical probability that you will not randomly choose a blueberry muffin?

A miss is the complement of a hit.

∴ The probability is 83%.

Not choosing blueberry is the complement of choosing blueberry.

∴ The probability is about 66.7%.

Homework: p. 218-220 #1, 3, 5, 6, 7, 8, 10, 12, 13