Probability Dual Event
In this lesson, you will learn how to find the probability of two events happening together. We will explore how to identify whether events are independent or dependent, and how that affects whether to use multiplication or addition.
Related Lessons:
Possibility Tree
A probability tree is a visual tool used to calculate the probability of multiple events happening in sequence. It divides each event into branches, showing all possible outcomes at every stage. Key points to remember when using a probability tree:
- Always check whether there is replacement or no replacement.
- If there is replacement, the denominator stays the same; if there is no replacement, the denominator decreases by 1.
Calculate Probability Using Possibility Tree
When using a probability tree to perform calculation, keep in mind:
- Events along the same branch are dependent, so multiply their probabilities.
- Events on different branches are mutually exclusive, so add their probabilities.
Probability Without Probability Tree
Using a probability tree can become very cumbersome in complex situations, such as drawing from more than two coloured balls. The tree quickly grows large and is very hard to manage.
At higher levels of study, we rely on mathematical reasoning to calculate the probability of multiple events.
One key point in probability notation is that we don’t explicitly show whether there is replacement or not.
For example, if we want to find the probability of drawing a red ball followed by a blue ball, we simply write it as P(R,B). Whether there is replacement or not is accounted for in the calculation.
Example 1
In a wardrobe, there are 9 blue shirts, 3 white shirts and 5 red shirts. Two shirts are taken out of the wardrobe at random and not replaced.
- Find, as a fraction in its simplest form, the probability that both shirts are of different colours.
A third shirt is drawn out. Find, as a fraction in its simplest form, the probability that
- none of the shirts are white.
- at least one of the shirt is white.
Example 2
At a hospital, the probability of a patient infected by the flu virus is \(\)\frac{2}{10}[\latex]. The probability that an infected person will be diagnosed correctly as carrying the virus is \(\)\frac{9}{10}[\latex]. The probability that a non-infected person will be diagnosed wrongly is \(\)\frac{1}{100}[\latex].
- Complete the probability diagram shown in the answer space.
- A patient is chosen at random from the hospital. Find the probability that he will be diagnosed wrongly.
Example 3
- 2 marbles are white and black in colour.
- 2 marbles have the same colour.
- 2 marbles have different colours