**List all outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays. Assign probabilities to outcomes and determine probabilities for events (ACMSP225)**

**LO: To construct a two-step experiment
**

**Know:**

- How to construct an experiment

**Understand:**

- that sample spaces can be constructed for two-step experiments

**Do:**

- I can construct a sample space for a two-step chance experiment.

**Independent Events**

**Independent Events**

### Independent events are events that are not affected by previous events.

### For example, if you flip a coin, it has a probability of 0.5 for landing heads.

### To get the probability of landing 2 heads, you just multiply the probability of landing heads twice. (0.5 x 0.5) = 0.25 or 1/4.

### To get the probability of landing 3 heads, you just multiply the probability of landing heads three times. (0.5 x 0.5 x 0.5) = 0.125 or 1/8.

.

*Tree Diagrams*

### Tree diagrams can also be used to help figure out the probability of events.

### Tree diagrams basically help you visualise all the different possible combinations of events which can occur.

### If you look at the tree diagram, there is only 1 (HHH) out of a possible 8 outcomes, hence the probability of landing 3 consecutive heads is 1/8.

.

*Dependent Events*

*Dependent Events*

### Dependent events are events that are affected by previous events.

### For example, if you take a marble out of a bag, the probability changes based on the results of the first result.

### The probability of drawing a blue marble in the first instance is 2 out of 5 or (0.4). Now there can be two possible outcomes, you could have drawn a red marble, which would then give you a 2 in 4 chance of drawing a blue marble in the second instance. Or you could have drawn a blue marble in the first instance which would leave you with a 1 in 4 chance of drawing a blue marble in the second instance.

*Tree Diagrams*

Tree diagrams can once again be used to help figure out the probability of events.

*Tree Diagrams*

### What are the chances of drawing 2 blue marbles? There’s a 2/5 chance (0.4) in the first instance, and a 1/4 (0.25) in the second instance. Once again we multiply the probabilities 1/10 (0.1) or 10% chance of drawing 2 blue marbles.

**Student Work**

**Student Work**

**ACARA Work Samples**

**ACARA Work Samples**

**Probability Videos**

**Probability Videos**