The chance of a happening is named as the probability of the event happening. It tells us how likely an occasion is going to happen; it doesn’t tell us what’s happening. There is a fair chance of it happening (happening/not happening). They’ll be written as decimals or fractions. The probability of occurrence A is below.
P (A) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of total possible outcomes}}\)
Following are two varieties of probability:
- Experimental probability
- Theoretical probability
What is Experimental Probability
Definition: Probability that’s supported by repeated trials of an experiment is named as experimental probability.
P (event) = \(\frac{\text{Number of times that event occurs}}{\text{Total number of trails}}\)
Example: The table shows the results of spinning a penny 62 times. What’s the probability of spinning heads?
23 | 39 |
Heads | Tails |
Solution: Heads were spun 23 times in a total of 23 + 39 = 62 spins.
P (heads) = \(\frac{\text{23}}{\text{69}}\) = 0.37 or 37.09 %
Definition: When all possible outcomes are equally likely the theoretical possibility of an incident is that the quotient of the number of favorable outcomes and therefore the number of possible outcomes.
P (event) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\)
Example: You randomly choose one among the letters shown. What’s the theoretical probability of randomly choosing an X?
Solution: P (x) = \(\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\) = \(\frac{\text{1}}{\text{7}}\) or 14.28%
A prediction could be a reasonable guess about what is going to happen in the future. Good predictions should be supported by facts and probability.
Predictions supported theoretical probability. These are the foremost reliable varieties of predictions, based on physical relationships that are easy to work and measure which don’t change over time. They include such things as:
- coin flips
- spinners
- number cubes
Let’s take a look at some differences between experimental and theoretical probability:
Experimental probability | Theoretical probability |
|---|---|
Experimental probability relies on the information which is obtained after an experiment is administered. | Theoretical probability relies on what’s expected to happen in an experiment, without actually conducting it. |
Experimental probability is that the results of the quantity of occurrences of a happening / the whole number of trials | Theoretical probability is that the results of the quantity of favorable outcomes / the entire number of possible outcomes |
Example: A coin is tossed 10 times. It’s recorded that heads occurred 6 times and tails occurred 4 times. P(heads) = \(\frac{6}{10}\) = \(\frac{3}{5}\) P(tails) = \(\frac{4}{10}\) = \(\frac{2}{5}\) | Example: A coin is tossed. P(heads) = \(\frac{1}{2}\)
P(tails) = \(\frac{1}{2}\) |
1. What is the probability of tossing a variety cube and having it come up as a two or a three?
Solution:
First, find the full number of outcomes
Outcomes: 1, 2, 3, 4, 5, and 6
Total outcomes = 6
Next, find the quantity of favorable outcomes.
Favorable outcomes:
Getting a 2 or a 3 = 2 favorable outcomes
Then, find the ratio of favorable outcomes to total outcomes.
P (Event) = Number of favorable outcomes : total number of outcomes
P (2 or 3) = 2:6
P (2 or 3) = 1:3
The solution is 1:3
The theoretical probability of rolling a 2 or a 3 on a variety of cube is 1:3.
2. A bag contains 25 marbles. You randomly draw a marble from the bag, record its color, so replace it. The table shows the results after 11 draws. Predict the amount of red marbles within the bag.
COLOR | FREQUENCY |
|---|---|
Blue | 1 |
Green | 3 |
Red | 5 |
Yellow | 2 |
Solution:
To seek out the experimental probability of drawing a red marble.
P (EVENT) = \(\frac{\text{Number of times the event occurs}}{\text{Total number of trials}}\)
P (RED) = \(\frac{\text{5}}{\text{11}}\) (You draw red 5 times. You draw a complete of 11 marbles)
To make a prediction, multiply the probability of drawing red by the overall number of marbles within the bag.
\(\frac{\text{5}}{\text{11}}\) x 25 = 11.36 ~ 11 so you’ll be able to predict that there are 11 red balls in an exceedingly bag
3. A spinner was spun 1000 times and the frequency of outcomes was recorded as in the given table.
Outcome | Red | Orange | Purple | Yellow | Green |
|---|---|---|---|---|---|
Frequency | 185 | 195 | 210 | 206 | 204 |
Find (a) list the possible outcomes that you can see in the spinner (b) compare the probability of each outcome (c) find the ratio of each outcome to the total number of times that the spinner spun.
Solution:
(a) The possible outcomes are 5. They are red, orange, purple, yellow, and green. Here all the five colors occupy the same area in the spinner. They are all equally likely.
(b) Compute the probability of each event.
P (Red) = \(\frac{\text{Favorable outcomes of red}}{\text{Total number of possible outcomes}}\) = \(\frac{\text{1}}{\text{5}}\) = 0.2
Similarly, P (Orange), P (Purple), P (Yellow) and P (Green) are also \(\frac{\text{1}}{\text{5}}\) or 0.2.
(c) From the experiment the frequency was recorded in the table.
Ratio for red = \(\frac{\text{Number of outcomes of red in the above experiment}}{\text{Number of times the spinner was spun}}\) = \(\frac{\text{185}}{\text{1000}}\) = 0.185
Similarly, we can find the corresponding ratios for orange, purple, yellow, and green are 0.195, 0.210, 0.206, and 0.204 respectively. Can you see that each of the ratios is approximately equal to the probability which we have obtained in (b) [i.e. before conducting the experiment]
The experimental probability of an occurrence is predicted by actual experiments and therefore the recordings of the events. It’s adequate to the amount of times an incident occurred divided by the overall number of trials.
When all possible events or outcomes are equally likely to occur, the theoretical probability is found without collecting data from an experiment.
Experimental probability, also called Empirical probability, relies on actual experiments and adequate recordings of the happening of events. To work out the occurrence of any event, a series of actual experiments are conducted.
Theoretical probability describes how likely an occurrence is to occur. We all know that a coin is equally likely to land heads or tails, therefore the theoretical probability of getting heads is 1/2. Experimental probability describes how frequently a happening actually occurred in an experiment.
So the results of flipping a coin should be somewhere around 50% heads and 50% tails since that’s the theoretical probability.
No, since the quantity of trials during which the event can happen can not be negative and also the total number of trials is usually positive.