Probability of Combined Events

Conditional Probability

Probability of an event occurring given that another event has occurred.

For example:

Remember to consider the type of events and their relationships when calculating the probability of combined events.

Probability of Combined Events using Sample Space Diagrams

A sample space diagram is a visual representation of all possible outcomes of an experiment. It can be used to calculate the probability of combined events by identifying the favorable outcomes and dividing them by the total number of possible outcomes.

To calculate the probability of combined events using a sample space diagram:

1. Draw a sample space diagram showing all possible outcomes of the experiment.
2. Identify the favorable outcomes that meet the conditions of the combined event.
3. Count the number of favorable outcomes.
4. Divide the number of favorable outcomes by the total number of possible outcomes in the sample space.

Example:

What is the probability that both coin A and coin B land heads up if both coins are flipped simultaneously?

Solution:

Favorable Outcomes: HH (both coins land heads up)

Number of Favorable Outcomes: 1

Total Number of Possible Outcomes: 4

Probability: $1/4$ or 0.25

This means that the probability of both coins landing heads up is $1/4$ or $25\%$.

Probability of Combined Events using Diagram

There are several types of sample space diagram, including:

• Possibility Diagrams
• Tree Diagrams
• Venn Diagrams

Example:

Maria and Haleema toss 2 coins Haleema after Maria.

(a) Draw tree diagram of the experiment. Using tree diagram, find probabilities:

(b) Both tails up (c) Maria's coin head up and Haleema's coin tail up

Solution:

(a) Tree diagram

(b) Let E1 be the probability of both heads up, then obviously there is only one such outcome among total of 4 outcomes. i.e. E1 = {HH}

(c) Let E2 be the probability of Maria's coin head up and Haleema's coin tail up, then obviously there is only one such outcome among total of 4 outcomes. i.e. E2 = {HT}

(c) Probability of Combined Events using Venn Diagram

A Venn diagram is a pictorial representation of sets and their relationships, using overlapping /disjoint circles to represent the intersection of sets. It can be used to calculate the probability of combined events by identifying the regions that represent the favorable outcomes and calculating their probabilities.