What is Probability?
Probability indicates the chances of something happening. In our daily life we can observe such activities where we can visualize the concept of probability.
For example – if we flip a coin, we will talk about probability whether the flipped coin lands heads up or tails up. The measure of probability lies from 0 to 1 or sometimes in percentage from 0% to 100 %.
If an event has probability 0 then it means the event will not happen and probability of 1 means that the event will happen.
Sample space
Sample space is defined as the set of all possible outcomes of one or more events.
The total probability of all the outcomes of an event is equal to 1.
Sample point
Each outcome in a sample space is called a sample point of that sample space.
For example: When we roll a dice there are 6 outcomes that could happen, which are 1, 2, 3, 4, 5, 6. Therefore, the set of all these outcomes is called sample space and each outcome is called sample point.
Compound Events
If an event has more than one sample point, it is termed as a compound event.
For example, the probability of tossing a tail on a coin, then rolling an odd number on a die is a compound event.
Probability of Compound Event Formula
Below is the formula to calculate the probability of a compound event.
Probability of a compound event = \(\frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}\)
We will discuss this formula in detail in the next section.
How to use Probability of Compound Event Formula?
Probability of a compound event is the probability of 2 or more events occurring together. Probability of a compound event can be calculated by dividing the number of favorable outcomes in the event by the total number of outcomes.
Mathematically,
Probability of a compound event, P = \(\frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}\)
Steps to find out the probability of compound events:
- Find out the sample spaces for the events.
- Count the total number of possible outcomes of the events.
- Count the possible favorable outcomes from the trials.
- Calculate probability by using the formula.
Solved Examples
Example 1: What is the probability of rolling a number greater than 5 and flipping tails?
Solution:
Sample space of rolling a number and flipping a coin given as-
1H | 2H | 3H | 4H | 5H | 6H |
|---|---|---|---|---|---|
1T | 2T | 3T | 4T | 5T | 6T |
So, total number of possible outcomes = 12
There are only 1 favorable outcomes in the sample space for rolling a number greater than 5 and flipping tails that is: 6T
\(\text{P (event)}=\frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}\) [Probability of compound event formula]
\( =\frac{1}{12} \) [Substitute values]
Hence, the probability of rolling a number greater than 5 and flipping tails is \(=\frac{1}{12}\).
Example 2: David has 5 shirts and 2 jeans. How many different outfits he can make from these shirts and jeans.
Solution :
By using the fundamental counting principle, we will try to find the number of possible outcomes for each event.
So, choosing a shirt has 5 possible outcomes.
And choosing jeans has 2 possible outcomes.
So, by applying the fundamental counting principle, the total number of different outfits he can make from shirt and jeans
= 5 × 2 [Multiply]
= 10
Hence, David can make 10 different outfits.
Example 3: What is the probability of getting heads by tossing a coin twice.
Solution :
Sample space of flipping a coin twice given as-
HH | TH | HT | TT |
|---|
Therefore, the total number of possible outcomes = 4
Total number of favorable outcomes = 3 (HH, HT, TH)
Now, P (event) = \(\frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}\)
= \(\frac{3}{4}\) [Substitute]
Hence, the probability of getting heads by tossing a coin is 34.
Example 4 : A train has one engine and four train cars. Find the total number of ways an engineer can arrange the train.
Solution:
Since, there are 4 cars. So the first car could be any of the 4. The next car could be any of the remaining 3. The next could be any of the remaining 2. The next will be remaining 1.
So, the total number of ways an engineer can arrange the train
= 4 × 3 × 2 × 1
= 24 [Multiply]
Hence, there are 24 ways an engineer can arrange the train.
Example 5 : A dice is thrown 100 times and outcomes are noted as given below:
Outcome | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
Frequency | 20 | 8 | 15 | 22 | 19 | 16 |
If a die is thrown at random, find the probability of getting 2.
Solution:
Given total number of trials = 100
From the table, number of times 2 comes up = 8
Probability of getting 2 = \(\frac{\text{frequency of 2}}{\text{total number of trials}}\)
= \(\frac{8}{100}\) [Substitute]
= \(\frac{2}{25}\) [Divide numerator and denominator by 4]
Hence, the probability of getting 2 is \(\frac{2}{25}\).
If an event has more than one sample point, it is termed as a compound event. The compound events are a little more complex than simple events. These events involve the probability of more than one event occurring together while simple events involve probability of only one event.
Experimental probability is defined as the probability that is based on repeated trials of an experiment.
Theoretical probability is defined as the ratio of the possible number of favorable outcomes and the total number of Outcomes of an event. In theoretical probability all possible outcomes are equally likely.
According to the fundamental counting principle if an event A has “m” possible outcomes, and another event B has n possible outcomes then the total number of outcomes of event A followed by event B is m\(\times\)n.
A compound event can have one or more than one outcome.
For example – The probability of rolling an even number on a dice then tossing a head on a coin has more than one outcome.
The probability of flipping a coin twice to get heads on both coins has only one possible outcome.