Probability Diagrams

Probability Diagrams

Possibility diagram and tree diagram

Probability Diagram

When an event or experiment is more complex, probability diagrams are used to display all the possible outcomes of an event.

Possibility Diagram

Possibility Diagram

A black dice and a white dice are thrown at the same time. Display all the possible outcomes. Find the probability of obtaining

a) a total of 5

b) a total of 11

c) a '2' on a black dice and a '6' on a white dice.

It is convenient to display all the possible outcomes on a grid.

Possibility Diagram

There are 36 possible outcomes, shown where the lines cross.

a) There are 4 ways of obtaining a total of 5 on the two dice. They are shown circled on the diagram.

The probability of obtaining a 5 is 4/36

Possibility Diagram

There are 36 possible outcomes, shown where the lines cross.

b) There are 2 ways of obtaining a total of 11 on the two dice. They are shown with a cross on the diagram.

P_{(total of 11)} = 2/36 = 1/18

Possibility Diagram

There are 36 possible outcomes, shown where the lines cross.

c) There is only 1 way of obtaining a '2' on a black dice and a '6' on a white dice.

P_{(2 on black & 6 on white)} = 1/36

Possibility Diagram

The diagram shows 2 spinners, each of which is divided into 4 equal sectors. Each spinner has a pointer which, when spun, is equally likely to become to rest in any of the four equal sectors.

In a game, each pointer is spun once. Find the probability that

a) the pointers will stop at the same number

b) the first spinner shows the larger number

Possibility Diagram

Construct a possibility diagram. Each dot represents a possible outcome according to the coordinates.

a) the pointers will stop at the same number

Let A = event of getting the same number on the 2 spinners

P_A = 4/16 = 1/4

Possibility Diagram

Construct a possibility diagram. Each dot represents a possible outcome according to the coordinates.

b) the first spinner shows the larger number

Let B = event the first spinner shows the larger number

P_B = 6/16 = 3/8

Multiple Choice

There are 4 possible outcomes, find the probability of obtaining 2 heads

• $\frac{2}{4}$

• $\frac{1}{4}$

• $\frac{1}{2}$

• $\frac{3}{4}$

Multiple Choice

What is the sample space when you roll a die and then flip a coin?

• 1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T

• 2

• 1,2,3,4,5,6,H,T

• 12

Tree Diagram

Tree Diagram

A tree diagram is a branching structure that maps the outcomes of chances/probabilities. The more branches, the more outcomes.

Tree Diagram

Flipping a coin. Your only outcomes are either head or tail. Your tree diagram would look like this:

Tree Diagram

Flipping a coin twice. If you flipped your coin a second time, you still only have the outcomes heads and tails but this time they branched out from your previous outcome. So your tree diagram would look like this:

Tree Diagram

Tom and Sam take turns to throw a dart at a target

•The probability that Tom hits the target is 0.3

•The probability of Sam hits the target is 0.2

Please complete the tree diagram

Tree Diagram

What is the probability that Tom and Sam both hit the target?

P_{(H, H)} = 0.3 * 0.2 = 0.06

Tree Diagram

Michael throws a stone at a target. He then throws a second stone at the target.

•The probability he hits the target on his first throw is 1/5

•If he hits the target on his first throw, the probability he will hit the target on his second throw is 1/3.

•If he misses the target on his first throw, the probability he will hit the target on his second throw is ¼.

•The probability he hits the target on his first throw is 1/5

•If he hits the target on his first throw, the probability he will hit the target on his second throw is 1/3.

•If he misses the target on his first throw, the probability he will hit the target on his second throw is ¼

Please complete the tree diagram

Work out the probability that Michael will hit the target on both throws

P_{(Hit, Hit)} = 1/5 * 1/3 = 1/15

Open Ended

Work out the probability that Michael will miss the target on both throws