**Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence(ACMSP246)**

**LO:** To construct a two/three-step experiment and assign probabilities.

**Know:**

- How to construct an experiment
- The definition of with and without replacement
- Probability is a scale from 0 to 1.
- Determine the probability of events.
- The definition of independent and dependent events.

**Understand:**

- that the probability of events can be calculated due to it’s circumstances.

**Do:**

- I can construct a sample space for a two/three-step chance experiment and assign probabilities.

**Independent Events**

**Independent Events**

### Independent events are events that are not affected by previous events. For example, if I’m flipping a coin, it has nothing to do with the outcome of rolling a die. If I go to McDonalds and order a burger, it has no influence over what drink I choose to order.

### 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.

**Dependent Events**

**Dependent Events**

### Dependent events are opposite to independent events and that they 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 draw 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.

### So with dependent events, the outcome of the first event changes the chances or probability of the second event. Another way of thinking about it is that the probability or chance of the second event depends on the outcome of the first event.

**Tree Diagrams**

**Tree Diagrams**

### Tree diagrams can also be used to help figure out the probability of events because it can be used to map out all of the possible combinations for the outcomes of the events.

### 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.

### Tree diagrams can once again be used to help figure out the probability of events in both independent and dependent situations.

### 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.

**ACARA Work Samples**

**ACARA Work Samples**